B-Bounded cohomology and applications

B -BOUNDED COHOMOLOGY AN D APPLI CA TIONS RONGHUI JI, CRICHTON OGLE, A ND BOBBY W. R A MSEY Abstract. A discrete group with word-length ( G, L ) is B - isocohomological for a b ounding classes B if the comparison map fr om B -b ounded cohomology to ordinary cohomology (with coefficients in C ) i s an isomor phism; it is strongly B - isocohomological if the same is true with arbitrary coefficient s. In this pap er we establish some basic conditions guaran teeing strong B -iso cohomologica lity . In particular, w e sho w strong B - isocohomologicality f or an F P ∞ group G if all of the weigh ted G -sensitive Dehn functions are B -b ounde d. Suc h groups include all B -asynchronously combable groups; moreo v er, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. W e also provide examples where the comparison map fails to b e injectiv e, as well as surjectiv e, and gi ve an example of a solv able group with quadratic first Dehn function, but exp onen tial second Dehn function. Finally , a r el ative theory of B -bounded coh omology of groups with resp ect to subgroups is introduced. Relativ e iso cohomolog icality is dete rmined in terms of a new notion of r elative Dehn functions and a r elative F P ∞ property for groups with resp ect to a collection of subgroups. Applications for computing B -bounded cohomology of groups are given in the cont ext of r elativ ely h yperb olic gr oups and dev elopable complexes of groups. Contents 1. Int ro duction 2 2. Preliminarie s 6 2.1. Bounding Classes 6 2.2. The F P α and H F α conditions 7 2.3. B -homology a nd B -cohomolo gy of alg ebras 7 2.4. B -homology a nd B -cohomolo gy of weight ed c o mplexes 8 2.5. Dehn functions 10 2.6. Pro ducts, copr oducts and pa iring o p erations 15 3. B co homology of discrete groups 16 3.1. Com bable groups 16 3.2. B -iso cohomologicality a nd type B F P ∞ groups 20 4. Relative constructions 25 4.1. Relative H F n and the Brown-Bieri-E c k ma nn condition 25 4.2. Relative Dehn functions 29 4.3. Relative B -b ounded cohomo logy 31 5. Two sp ectral sequences in B -bo unded cohomolo gy 37 5.1. The Hochsc hild-Serre spectral sequence 37 5.2. The spectra l sequence asso ciated to a group a cting on a complex 39 6. Dualit y groups a nd the comparison map 42 6.1. Dualit y and Poincar´ e Dualit y Groups 42 6.2. Iso cohomologicality and the fundamental cla ss 42 1 2 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY 6.3. B -duality g roups 44 6.4. Tw o solvmanifolds 44 6.5. More on the compariso n map 47 References 49 1. Introduction T o a b ounding class of functions B , a group with length ( G, L ) and a weighted G -complex ( X , w ) o ne can asso ciate the B -bo unded, G -equiv ariant cohomology of X with coefficients in a H B ,L ( G )-mo dule V : B H ∗ G ( X ; V ) The constr uc tio n is a v ariant on that used in the no n-bounded case. Sta r ting with ( G, L ) and B , one constructs a b ornolo g ical algebra H B ,L ( G ) - a completion of the gr oup algebra C [ G ] - a nd a b ornologica l H B ,L ( G )-mo dule B C ∗ ( E G × X ) - a similar c o mpletion of the the complex of singular chains C ∗ ( E G × X ). Then giv en a bornolo gical H B ,L ( G )-mo dule V , one for ms the (co)complex Hom bdd H B ,L ( G ) ( B C ∗ ( E G × X ) , V ) of b ounded H B ,L ( G )-mo dule homomo rphisms. The cohomology groups B H ∗ G ( X ; V ) are then defined as the (algebraic) cohomology groups of this complex. Ther e is a natural tr ansformation o f functors B H ∗ − ( − ; − ) → H ∗ − ( − ; − ) which, for giv en v a lue s , is referred to as the compariso n map Φ ∗ B = Φ ∗ B ,G ( X ; V ) : B H ∗ G ( X ; V ) → H ∗ G ( X ; V ) One wan ts to know the prop erties of this map; when it is injective, surjective, and what structures are pres erv ed under it. In some sp ecial cases, quite a bit is kno wn. F or example, tak ing B = B min = { co nstan t functions } yields H B min ,L ( G ) = ℓ 1 ( G ), and the res ulting cohomology theo ry is simply the equiv ar ian t b ounded cohomolo gy of X with co efficients in the Banach ℓ 1 ( G )-module V . At the other extreme, when B = B max = { f : R + → R + | f non-decrea sing } , the compa r ison map bec o mes an isomor phism under appropriate finiteness conditions: G is an F P ∞ group and X is a G -co mplex with finitely man y o rbits in eac h dimension. More interesting, and a lso more subtle, is the case when the bo unding cla ss B lies b et ween these t wo extremes, beca use it is in this range that the weigh t function on X and w or d- length function on G have the p oten tial for influencing the B -bounded co ho mology groups in a non-triv ial way . T o illustrate why this is of interest, we consider tw o applications. • The topo logical K -theory of ℓ 1 ( G ). Here the b ounding cla ss B = P = { non- decreasing p olynomials } is o f particular in ter e st, a s H P ,L ( G ) is a s mooth subalgebra o f ℓ 1 ( G ). T aking X = pt and V = C , the image of Φ ∗ B con- sists precise ly of thos e cohomology classes in H ∗ ( G ) = H ∗ ( G ; C ) whic h are p olynomially b ounded with resp ect to the word-length function on G . Such cla sses therefor e sa tisfy the ℓ 1 -analog ue of the Novik ov Conjecture, B -BOUNDED COHOMOLOGY AND APPLICA TIONS 3 and there is reason to b eliev e they also satisfy the Strong Novik ov Co n- jecture (without an y additional Ra pid Decay condition on the gr oup). F or this application, one would lik e the comparison ma p to be surjective . • The Strong ℓ 1 -Bass Co njecture. Here one is interested in determining the image of the Chern character ch ∗ : K t ∗ ( ℓ 1 ( G )) → H C top ∗ ( H P ,L ( G )) [ 21 ]. In this case, the pro blem of showing that the image lies in the “elliptic summand” amounts to proving the injectivity of the compa rison map (for suitable c hoice of X ). Of course, bo th proper ties ho ld when Φ ∗ B is bijectiv e , i.e., when G or more pre- cisely ( G, L ) is iso cohomolo gical [ 25 , 26 ]. In fact, up until this p oin t, all pro ofs of ei- ther injectivit y or surjectivit y for a given b ounding class ha ve arisen b y a verification of this strong er iso cohomolo gical prop ert y . The first type of r esult in this direction is due to the firs t author, who s howed in [ 20 ] that H C top ∗ ( H P ,L ( G )) ∼ = H C ∗ ( C [ G ]) for groups o f polyno mial gr o wth. Subsequently , it was determined independently by the seco nd a uthor in [ 3 4 ] and R. Meyer in [ 25 ] that ( G, L ) is P -iso cohomolog ical in the case G admits a synchronous co m bing. Moreov er, in [ 34 ] it was shown that iso cohomologica lit y with a rbitrary co efficient s, or strong B -iso cohomolog ic alit y ,(at least for B = P ) follow ed for H F ∞ groups whenever all of the Dehn functions (as defined in [ 34 ]) were p olynomially b ounded. This la st res ult w a s significantly strengthened by the firs t and third authors in [ 23 ], wher e it was established (a gain for B = P ), that strong P -iso cohomolo gicalit y for a n F P ∞ group was actually equiv alent to the existence of p olynomially bounded Dehn functions in each de- gree. F r o m this the authors were able to conclude that the comparison map (with X = pt ), fails to be surjectiv e for rather simple groups when o ne allows non-trivial co efficien ts. How ever, still unknown for the sta ndard word-length function o n G and the po lynomial b ounding cla ss P , or more gener ally any b ounding class B containing the linear p olynomials L , w ere answers to the following questions: Q1. Is the compariso n map Φ ∗ B = Φ ∗ B ( G ) : B H ∗ ( G ) → H ∗ ( G ) always surjective? Q2. Is Φ ∗ B ( G ) alwa ys injective? Q3. If G is a n H F ∞ group, are the higher Dehn functions of G P -eq uiv a len t to the first Dehn function of G ? In fact, it w as c onjectured by the second author in [ 34 ] that the answer to the third question was “yes”, given that all k no wn examples at the time suggested this to b e the case. Nevertheless, one of the co nsequences of the results of this pap er is Theorem A The a nsw er to each of these questio ns is “no”. Precisely , we show A1. (following Gro mo v ) There ex is ts a compact, closed, orientable 3-dimensio nal solvmanifold M 3 1 ≃ K ( G 1 , 1 ) with a 2- dimensional cla s s t 2 ∈ H 2 ( M 1 ) = H 2 ( G 1 ) which is not B -b ounded for any B ≺ E , the b ounding class of simple exp onen tia l functions. A2. There exists a compac t, c losed, orientable 5-dimensional solvmanifold M 5 2 ≃ K ( G 2 , 1 ), where the first Dehn function of G 2 is quadratic, but the second Dehn function is at least simple exponential. A3. If B is a b ounding c la ss with B  L , and G is a finitely- presen ted F L g roup (meaning B G is homoto p y-e q uiv a len t to a finite complex) for which the 4 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY compariso n ma p Φ ∗ B is no t surjective with resp ect to the standar d word- length function (a s in (A1.)), there is another disc r ete g roup F ( G ) depe nd- ing on G “up to homo top y” for w hich the compar ison map is not injectiv e with re s pect to the sta ndard word-length function on F ( G ). Moreover, if B  P , then F ( G ) can b e tak en to b e als o of t yp e F L . Somewhat surpr ising is the sha rpness of these results. F or (A1.), this is the simplest type of finitely-presented gro up a nd smallest co ho mological dimension in which surjectivit y with trivia l coefficients can fail. In the ca se of (A2.), we note that a linear first Dehn function implies the group is hyperb olic, in tur n implying that all of the hig her Dehn functions a re als o linear . Moreov er , there is an isop erimetric gap that o ccurs be tw een degree one and tw o , so that if the first Dehn function is not linear, it m ust be at least qua dratic - the smallest degree for which the second Dehn function c o uld be non-p olynomial, or even non-linear. Finally , (A3.) provides, for e a c h bounding class B  P an injection from the set of (iso morphism classes of ) finitely-presented F L groups with non-surjective comparis on map to the set o f (isomorphism cla sses of ) finitely-presen ted F L gro ups with non-injective compariso n map. Up unt il no w, the main problem in studying either B -iso cohomologicality or strong B -iso cohomolo gicalit y has b een the abs e nc e of appro priate co mputatio nal to ols. The difficult y in extending classica l techniques lies in the analysis o f the group structures and the geometry of the asso ciated Cayley g raph. The primary goal of this pap er is to develop systematic metho ds for calcula ting B -b ounded co- homology , and to establis h iso cohomolog icalit y for a go o d class of gro ups. One such class c onsists of group extensio ns where the normal and quotient groups are iso co- homologica l with resp ect to the r estricted a nd quo tien t le ng th functions induced by the length function on the middle group. The o ther main cla s s of gr oups consider ed are those asso ciated with developable complexes o f groups; this inc ludes the c la ss of relatively hyper bolic g roups. The ma in c omputational tec hnique s in tro duced are the Ho chsc hild-Ser re sp ectral sequence in B - bounded cohomology asso ciated to an e x tension of g roups equipp ed with length functions, and the Serre sp ectral sequence in B -b ounded cohomology for developable co mplexes of g roups. As has bee n noted b y Meyer in [ 26 ], the ca teg ory in which one do es homological alg ebra in the bo rnological framework is almost never abe lia n, whic h ma k es the extension o f homologica l techniques from the non-b ounded to the bounded setting problematic. Among the res ults included below, w e hav e Theorem B Let G b e a finitely presented gr oup a cting cocompac tly on a con- tractible simplicial complex X without inv er s ion, with finitely ge ne r ated stabilizers G σ for each vertex σ in X , and with finite edg e stabilizers. Suppose a lso tha t X is equipp ed with the 1-skeleton weigh ting, and all of its higher weighted Dehn functions are B -b ounded. Then if each G σ is strongly B -iso cohomolog ical, G is as well. In [ 37 ] and [ 9 ] the notion of the first unw eig h ted ‘relative Dehn function’ is int ro duced for a g roup relative to a family of subgroups. This relative Dehn func- tion is well-defined for dev elo pa ble complexes of gro ups with finite e dge stabilizer s; in particular for rela tiv ely hyper bolic gr oups. In tuitively , the un w eighted relative Dehn functions b ound ‘relative fillings ’ o f relative cycles in a ‘relatively contractible space’. Th us , one exp ects that the compar ison map from B -b ounded relative co- homology to non-b ounded relative co homology is an isomorphism when the re la tiv e B -BOUNDED COHOMOLOGY AND APPLICA TIONS 5 Dehn functions are appropr ia tely bounded; i.e. the group is relatively is ocoho mological with respec t to the family of subgroups. Now it should be noted that the existence of Dehn functions, even in the abso lute setting, is not g uaranteed by the existence o f a nice r esolution. In genera l, one needs to w ork with weighted Dehn functions where the weigh ting degr eewise is either equiv alent to a w eighted ℓ 1 -norm asso ciated to a prop er weigh t function o n an additive set of genera tors, or is one over which there exists s ome geometric control. Using the 1-sk eleton weigh ting, we show Theorem C Suppo se the finitely pre s en ted gr oup G is F P ∞ relative to a finite family of finitely genera ted subgroups H . Then the following ar e equiv alent. (1) The weigh ted relative Dehn functions of E G rela tiv e to E H are B -b ounded. (2) G is strongly B -is ocoho mological re la tiv e to H . (3) The compariso n map B H ∗ ( G, H ; A ) → H ∗ ( G, H ; A ) is sur jectiv e for a ll bo rnological H B ,L ( G )-mo dules A . W e should als o r emark that, as in the ca se of relative group cohomolog y [ 4 , 7 ], there is a long-ex act sequence in B - b ounded cohomo logy r elating the b ounded c o ho- mologies of the subgr o ups and the group with the B -b ounded r elativ e cohomolo gy of the pair. An outline o f the pap er is as follows. In section 2, we recall fro m [ 21 ] some bas ic terminology regarding b ounding classes, a nd the setup fo r defining the G -e quiv ariant B -bo unded cohomology of bo rnological alge bras and weighted complexes. W e also define what we mean b y a Dehn function in this pape r . F or F P ∞ groups, ther e are a num ber o f differ- ent wa y s for defining Dehn functions, the most natural being a lgebraically defined Dehn functions whic h take into account the action of the group G . Also in this section we construct s ome basic pairing op erations b etw een B -b ounded homolog y and cohomology that are used la ter on. In section 3, we show 1 that asynchronously combable groups are type H F ∞ , via an explicit coning argument that allows us in section 3.2 to show tha t if the combing lengths ar e B -b ounded, so ar e a ll of the Dehn functions of G . W e also extend the main result of [ 23 ] to ar bitrary b ounding classes B . In section 4, we generalize the co nstructions and results o f section 3 to the rel- ative context. W e b egin b y establishing a relative version o f the Br o wn and Bieri- Eckmann conditions use d to establish homologica l or homo topic finiteness through a g iven degree. In section 4 .2, we introduce the higher dimensio nal relative Dehn functions in several differ en t, but equiv alent wa y s. In section 4.3 the no tion of relative B - b ounded cohomolog y is develop e d a nd sho wn to fit into a long-ex act sequence similar to the long - exact sequence in non-b ounded rela tiv e group coho- mology . The notion of relative B -iso cohomologica lit y is int ro duced and rela ted to the higher relative Dehn functions. This relationship is then examined in the case of r elativ ely hyperb olic groups and gr oups acting on complexes. In sec tion 5, w e co ns truct in B H ∗ ( · ) the i) Ho chsc hild-Ser re s pectral sequence asso ciated to a n extension of gro ups with word-length, and ii) the sp ectral sequence asso ciated to a group acting on a complex . These spe c tral sequenc e s closely resem- ble their non-b ounded counter-parts, and, as we men tioned ab ov e, provide the ma in to ols for computing B -bounded cohomolog y . 1 This result has als o recen tly b een obtained in [ 8 ], using more geometric techniques. 6 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Finally , in section 6, we examine sp ecific examples, b eginning with duality groups (in the sense of B ieri-Eckmann). A striking fact, prov ed in section 6.2, is that B -iso cohomologica lit y fo r an o rien ted n -dimensio nal Poincare Dualit y group G is guaranteed by the B -b oundedness of a single cohomolo gy clas s in H n ( G × G ) coming from the restriction o f the Thom class for the diagonal embedding, i.e., the “dual” fundamen tal cla ss of G . F o r fundamen tal gr oups of co mpact oriented manifolds with connected b oundary , B -iso cohomolo g icalit y is guara n teed b y a B -b ound o n t wo separate cohomolo gy classes . In section 6.3, we in tr oduce the no tion o f a B -duality group, and show that when the fundamental ho mology clas s in B -b ounded homolo gy is in the imag e of the homology comparison map, the cohomology comparison map is injectiv e for all co efficien ts. Finally , in sec tio ns 6.4 and 6.5, we discuss the exa mples men tioned in (A1.) - (A3.) above. The second author would like to thank Ian Leary for an illuminating r emark regar ding [ 24 ]. The first and thir d authors a re grateful to Denis Os in for his communications relating to complexes of gr oups, relativ e Dehn functions, and the Arzhantsev a-Osin example o f an ex ponential growth solv able group that has qua- dratic first Dehn function, [ 3 ]. The authors w ould also like to thank the referee for their careful rea ding of this paper and for their helpful suggestio ns. 2. Preliminaries W e disc us s some co nstructions and termino lo gy that will b e used throughout the pap er. 2.1. Bounding Classes. Let S denote the set of non-decreasing functions { f : R + → R + } . Suppos e φ : S n → S is a function of sets, and B ⊂ S . W e will say that B is w eakly closed under the op eration φ if for e a c h n -tuple ( f 1 , . . . , f n ) ∈ B n , there exists an f ∈ B with φ ( f 1 , f 2 , . . . , f n ) < f . A bounding class then is a subset of B ⊂ S satisfying ( B C 1 ) it contains the consta nt function 1, ( B C 2 ) it is weakly closed under the ope rations of taking p ositive rational linear combinations ( B C 3 ) it is weakly closed under the op eration ( f , g ) 7→ f ◦ g for g ∈ L , whe r e L denotes the linear b ounding class { f ( x ) = ax + b | a, b ∈ Q + } . Naturally o ccurring classes b esides L a r e B min = { Q + } , P = the s e t o f p olynomials with no n-negative rational co efficien ts, the set E = { e f , f ∈ L} , and B max = S . A b ounding class is multiplicativ e if it is weakly c lo sed under multiplication, and comp osable if it is weakly closed under comp osition. More gener ally , given b ounding classes B and B ′ , we say B is a left resp. right B ′ -class if B is weakly closed under left resp. right comp o sition with elements of B ′ (th us, for example, all b ounding classes are right L -cla sses by ( B C 3)). Basic prop erties of bounding classes w ere discussed in [ 21 ]; for tec hnical reaso ns only compos able b ounding classes w ere considered in that pap e r, how ever, all of the results of of [JOR1, § 1.2] apply for this larger collec tio n o f cla sses. W e write B ′  B if every f ∈ B ′ is bounded ab ov e b y some elemen t f ∈ B , with equiv alence B ′ ∼ B if B ′  B and B  B ′ . Finally , B ′ ≺ B if B ′  B but B ′ is not equiv a len t to B . A bounding c lass B is w eakly co un table if there is a countable b ounding class B ′ with B ∼ B ′ . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 7 Con ven tion 1 . All b ounding classes c onsider e d wil l b e assume d we akly c oun table. Since equiv alent bounding classe s yield iso morphic results in what follows, this amounts to w o rking with countable b ounding clas ses. Given a weighted set ( X , w ) and f ∈ S , the seminor m | | f on Hom( X , C ) is g iven by | φ | f := P x ∈ X | φ ( x ) | f ( w ( x )). Then H B ,w ( X ) = { f : X → C | | φ | f < ∞ ∀ f ∈ B } . The most imp ortant fea ture of H B ,w ( X ) is that it is a n algebra whenever X is a semi-g roup and w is sub-additive with r espect to the m ultiplication on X . If X has a unit, then so do es H B ,w ( X ). W e will mainly be concerned with the cas e ( X , w ) = ( G, L ) is a discr ete group equipped with a leng th function L (meaning a function L : G → R + subadditive with res p ect to the m ultiplica tion on G , and in v aria n t under the inv o lution g 7→ g − 1 ). The length function L is called a word-length function (with resp ect to a generating set S ) if L (1 ) = 1 and there is a function φ : S → R + with L ( g ) = min ( n X i =1 φ ( x i ) | x i ∈ S, x 1 x 2 . . . x n = g ) When the gener ating set S is finite, ta k ing φ = 1 pro duces the standar d word-length function on G . 2.2. The F P α and H F α conditions. In this pape r the term complex will refer either to a s implicia l complex, p olyhedral complex, or s implicia l set. F or a complex X , we say X is type H F α ( α ≤ ∞ ) if | X | ≃ | Y | wher e Y is a C W complex with finitely man y cells through dimensio n α . This notion clear ly defines an equiv a- lence relation on the appropriate category of complexes, and admits an equiv aria n t formulation: for a discrete group G which acts either cellula rly or simplicially , a G -complex X is t yp e G − H F α if there is a strong G -ho motop y equiv alence X ≃ Y with Y ha ving finitely many G -orbits through dimension α . A group G is type H F α if its classifying space B G is type H F α , or e quiv alen tly , if E G is t yp e G − H F α . When α is finite a nd X is a simplicial complex r esp. simplicial set resp. poly- hedral complex, the H F α condition is equiv alent to saying X ≃ Y a simplicial complex resp. simplicia l set resp. po lyhedral complex with Y ( α ) finite. When α = ∞ and X is a polyhedra l co mplex, then X is type H F ∞ iff X ≃ Y a p olyhe- dral complex with Y ( n ) finite for all n < ∞ . Ho w ever, if X is either a simplicial s e t or simplicia l co mplex , the H F ∞ condition is e quiv alen t to the w e a k er statemen t X ≃ Y = lim − → Y n a direct limit of simplicial se ts resp. simplicial subc o mplexes, with the inclusion Y n ֒ → Y inducing an n -co nnected map of s paces | Y n | ֒ → | Y | . F or discr ete gr oups, the c o ndition H F α is equiv alent to req uiring tha t G is finitely pres e nted and type F P α . The standard F P α condition - that Z admits a resolution ov er Z [ G ] whic h is finitely-genera ted pro jectiv e through dimensio n α - is now kno w n to b e strictly w eaker than requiring G to b e H F α when G is not finitely- presented. Because the fr amew o rk used in this pap er for defining Dehn functions is algebraic, our primary fo cus will be on discrete groups of t ype F P α . Again, we remind the reader that F P α is eq uiv a len t to F F α - the condition that Z admits a r esolution over Z [ G ] b y free Z [ G ]-modules which which are finitely-genera ted through dimensio n α 2.3. B -homol o gy and B -cohomolo gy of algebras. There ar e a num b er of differ- ent settings in whic h one ca n develop the theor y of B -bo unded cohomo lo gy for non- discrete algebras. F or arbitrar y B , the mo s t natural is the bor nological framework 8 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY int ro duced by [ 26 , 2 7 ]. Giv en a bor nological algebr a A and b ornologic a l A -mo dules M and N , the derived functors T or A ∗ ( M , N ) and Ext ∗ A ( M , N ) ar e computed in the bo rnological category using standar d co nstructions from homologica l algebr a, with the constraint that pro jective or injectiv e r esolutions used are contractible v ia a bo unded linea r contraction. Before defining homo logy and cohomology , we wan t to po in t out that in b oth cases there is a “bo rnologically correct” r educed theory and an “algebraically cor- rect” unreduced theory . In general, given a chain co mplex ( C ∗ , d ∗ ), or a co complex ( D ∗ , d ∗ ), of b ornologica l vector spaces with b ounded differential, w e distinguish betw een the alg ebraic (co)homolo gy groups H a ∗ ( C ∗ ) := { H n ( C ∗ ) = ker( d n ) / im( d n +1 ) } , H ∗ a ( D ∗ ) := { H n ( D ∗ ) = ker( d n ) / im( d n − 1 ) } and the b ornologic al (co )homology gr oups H b ∗ ( C ∗ ) := { H n ( C ∗ ) = ker( d n ) / im( d n +1 ) } , H ∗ b ( D ∗ ) := { H n ( D ∗ ) = ker( d n ) / im( d n − 1 ) } Thu s given a b ornologica l a lg ebra, A , with unit, bo rnological A -mo dules M 1 , M 2 , pro jectiv e resolutions P i • of M i ov er A , and a n injective reso lution Q • of M 2 ov er A T or A,x ∗ ( M 1 , M 2 ) := H x ∗ ( P 1 • b ⊗ A M 2 ) = H x ∗ ( M 1 b ⊗ A P 2 • ) , x = a, b (1) Ext ∗ A,x ( M 1 , M 2 ) := H ∗ x (Hom bdd A ( P 1 • , M 2 )) = H ∗ x (Hom bdd A ( M 1 , Q • )) , x = a, b (2) Here Hom bdd A ( − ) denotes (in each degree) the bo rnological v ector spa ce o f b ounded A -mo dule homomor phisms. 2.4. B -homol o gy and B -cohomo logy of w e igh ted complexes. A weigh t function on a s et S is a map of se ts w : S → R + . Fix a weigh ted set ( S, w ), and write C [ S ] for the vector spac e ov er C with ba sis S . F or a b ounding class B , we may define seminorms on C [ S ] by      X s ∈ S α s s      f := X s ∈ S | α s | f ( w ( s )) , f ∈ B If ( G, L ) is a discrete group with length function L , a w eighted G -set is a weight ed set ( S, w ) with a G -action o n S satisfying (3) w ( g s ) ≤ C · L ( g ) + w ( s ) , ∀ g ∈ G, s ∈ S Let H B ,w ( S ) denote the completion o f C [ S ] with resp ect to the seminorms in (3). Then H B ,w ( S ) may b e view ed a s a b ornolog ical vector space, which is F rechet if there exists a coun table b ounding class B ′ with B ∼ B ′ . Note that C [ S ] is a mo dule ov er C [ G ] in the usual w ay: ( P λ i g i )( P β j s j ) = P i,j λ i β j g i s j . Prop osition 1. The mo dule st ructur e of C [ S ] ove r the gr oup algebr a C [ G ] ext en ds to a b ounde d b ornolo gic al H B ,L ( G ) -mo dule s t ructur e on H B ,w ( S ) . Pr o of. This follows b y the sa me estimates as those used to show H B ,L ( G ) is an algebra:     X λ g 1 g 1   X λ g 2 s 2     f = X s ∈ S      X g 1 s 2 = s λ g 1 λ s 2      f ( w ( s )) B -BOUNDED COHOMOLOGY AND APPLICA TIONS 9 ≤ X s ∈ S X g 1 s 2 = s | λ g 1 λ s 2 | f ( L ( g 1 ) + w ( s 2 )) ! ≤ X s ∈ S   X g 1 s 2 = s,L ( g 1 ) ≤ w ( s 2 ) | λ g 1 λ s 2 | f (2 w ( s 2 ))   + X s ∈ S   X g 1 s 2 = s,w ( s 2 ) ≤ L ( g 1 ) | λ g 1 λ s 2 | f (2 L ( g 1 ))   ≤    X λ g 1 g 1    1    X λ s 2 s 2    f 2 +    X λ g 1 g 1    f 2    X λ s 2 s 2    1 < ∞ where | − | 1 denotes the ℓ 1 -norm, and f 2 ∈ B is any function satisfying f (2 x ) ≤ f 2 ( x ) ∀ x .  Suppo se w and w ′ are t wo weight functions on a se t S . W e write w  w ′ if there exist positive c onstan ts A and B with w ( s ) ≤ Aw ′ ( s ) + B for all s ∈ S . The weight functions a re linea rly equiv ale nt if w  w ′ and w ′  w . If w and w ′ are linearly equiv alent weigh t functions on S , then H B ,w ( S ) = H B ,w ′ ( S ) for any b ounding class B . A weight ed simplicial set , r espectively weighted simplicial complex, ( X , w ) is a simplicial set, resp ectiv ely a simplicial complex, X = { X n } n ≥ 0 together with weigh t functions w n : X n → R + such that fo r each n and n -simplex σ , w n − 1 ( σ ′ ) ≤ w n ( σ ) when σ ′ is a face of σ , and (in the case o f simplicial sets) w n +1 ( s j ( σ )) = w n ( σ ) where s j represents a degeneracy map 2 . In b oth cases, we will simply refer to ( X, w ) as a w eighted complex. Given a discrete group with length ( G, L ), a weigh ted G -complex is a G -c o mplex weigh ted in s uc h a way tha t for each n , the action of G on ( X n , w n ) satisfies equation ( 3 ) ab ov e. In this case, completing C ∗ ( X ) degreewise produce s a bor no logical chain complex B C ∗ ( X ) := { H B ,w n ( X n ) , d n } n ≥ 0 When the action of G on X is free, the definition of the G -equiv ariant B -b ounded cohomolog y o f X with co efficien ts in a b ornolog ical H B ,L ( G )-module A is B H ∗ G,x ( X ; A ) := H ∗ x (Hom cont H B ,L ( G ) ( B C ∗ ( X ) ∗ , A )) , x = a, b (4) B H G,x ∗ ( X ; A ) := H x ∗ ( B C ∗ ( X ) b ⊗ H B ,L ( G ) A ) , x = a , b (5) Note tha t Hom bdd H B ,L ( G ) ( B C ∗ ( X ) ∗ , A ) = Hom bdd ( B C ∗ ( X ) ∗ , A ) G , and B C ∗ ( X ) b ⊗ H B ,L ( G ) A ident ifies with the quotient of B C ∗ ( X ) b ⊗ A by the c lo sure of the image of { I d − g | g ∈ G } w he r e g ◦ ( x ⊗ y ) = ( g x ⊗ g − 1 y ). In genera l when the action of G is not free, the definition is adjusted in the usua l way by first replacing these G -fixed-po int and G -orbit spaces by the la rger equiv ariant“homotop y fix ed-point” and “homotopy- orbit” spaces . Let E G deno te the homog eneous bar resolution of G , with weigh t function w ( g 0 , . . . , g n ) = L ( g 0 ) + P n i =1 L ( g − 1 i − 1 g i ). Then 3 Hom bdd ( B C ∗ ( X ) ∗ , A ) hG := Hom bdd H B ,L ( G ) ( B C ∗ ( E G ) , Hom bdd ( B C ∗ ( X ) ∗ , A )) ∼ = Hom bdd H B ,L ( G ) ( B C ∗ ( E G ) b ⊗B C ∗ ( X ) ∗ , A ) 2 The same definition applies to p olyhedral complexes, under a mil d r estriction on the num b er of faces allo we d i n each dimension. 3 the isomorphism of complexes B C ∗ ( E G ) b ⊗B C ∗ ( X ) ∗ ∼ = B C ∗ ( E G × X ) ∗ is a special case of a more general equiv alence to b e established in [ 22 ]. 10 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY ∼ = Hom bdd H B ,L ( G ) ( B C ∗ ( E G × X ) ∗ , A ); ( B C ∗ ( X ) b ⊗ A ) hG := ( B C ∗ ( E G ) b ⊗B C ∗ ( X ) ∗ ) b ⊗ H B ,L ( G ) A ∼ = B C ∗ ( E G × X ) ∗ b ⊗ H B ,L ( G ) A The G -equiv ariant B -b o unded cohomology groups of the w eighted complex ( X , w ) with coefficients in the H B ,L ( G )-module A are given as B H ∗ G,x ( X ; A ) := H ∗ x (Hom bdd ( B C ∗ ( X ) ∗ , A ) hG ) , x = a, b B H G,x ∗ ( X ; A ) := H x ∗ ( B C ∗ ( X ) b ⊗ A ) hG ) , x = a, b When the action of G o n X is free, these gr oups agr ee with those defined above. They also agree with those given in the previous section in terms of derived functors; they are simply eq ualities (1) and (2) wher e M 1 is the (DG) H B ,L ( G )-module B C ∗ ( X ). In this context, B H ∗ G,x ( X ; A ) = E xt ∗ H B ,L ( G ) ,x ( B C ∗ ( X ) , A ) (6) B H G,x ∗ ( X ; A ) = T or H B ,L ( G ) ,x ∗ ( B C ∗ ( X ) , A ) (7) with B C ∗ ( E G × X ) used as a ca nonical free resolution of B C ∗ ( X ) o ver H B ,L ( G ) when B C ∗ ( X ) is not fre e over H B ,L ( G ) (i.e., when the ac tio n of G on X is not free). the inclusion of complexes C ∗ ( E G × X ) ֒ → B C ∗ ( E G × X ) induces comparison maps Φ ∗ B : B H ∗ G,x ( X ; A ) → H ∗ G ( X ; A ) (8) Φ B x : H G ∗ ( X ; A ) → B H G,x ∗ ( X ; A ) (9) which are clearly functor ia l in X , G , and A . The B -b ounded cohomolo gy groups of ( X , w ) ar e computed as the coho mology of a subcomplex of C ∗ ( E G × X ; A ) G which ca n b e difficult to describ e in general. How ever, when the action of G is free o n X and trivial o n A , and A is simply a normed v e c tor space (e.g., C ), then B H ∗ G,x ( X ; A ) = B H ∗ x ( X/G ; A ) = H ∗ x ( B C ∗ ( X/G ; A )) where B C ∗ ( X/G ; A ) = {B C n ( X/G ; A ) } n ≥ 0 (10) B C n ( X/G ; A ) = { φ : ( X/G ) n → A | ∃ f ∈ B s.t. | φ ( x ) | < f ( w ( x )) ∀ x ∈ ( X/G ) n } (11) Con ven tion 2. Unless otherwise indic ate d, B H ∗ ( − ) wil l me an B H ∗ a ( − ) ; mor e gen- er al ly, B H ∗ G ( − ) wil l me an B H ∗ G,a ( − ) . 2.5. Dehn functions. There a re t wo basic environmen ts in which one can consider (higher) Dehn functions. W e discuss b oth. B -BOUNDED COHOMOLOGY AND APPLICA TIONS 11 2.5.1. The ge ometric setting. Supp ose X is a weakly con tractible complex, with bo undary map ∂ . Any lo op α in X (1) bo unds a disk β in X (2) , ∂ β = α . Denote the nu mber of n -cells in a complex W by k W k n . Set V ol 2 ( α ) = min k β k 2 , where this minim um is taken ov er all disks β in X (2) with ∂ β = α . More generally , if f : α → X is a mapping o f a combinatorial n - sphere to X , there is a map of an ( n + 1)-ball h : β → X ( n +1) with ∂ h = f , a nd the filling volume of f is V ol n +1 ( f ) = min k β k n +1 , where this minimum is tak en ov er all com binatorial ma ps of ( n + 1)-balls h : β → X with b oundary f . F or each n , the n t h geometric Dehn function of X , d n X : N → N , is defined via the form ula d n X ( k ) := max V ol n +1 ( f ) where the maximum is taken ov er a ll combinatorial maps f of n -spheres f : α → X with k f k n ≤ k . I n the case n = 1, d 1 X is often referred to as simply ‘the’ geo metric Dehn function of the complex. Thes e Dehn functions give a mea suremen t of the filling volume of cy c le s in X , with d n X being the n th un weigh ted Dehn function of X . O f course, these functions do not exis t if the cor respo nding maximum v alues do not exis t. When X comes eq uipped with a weigh t w on its cells, there is definition of geometric Dehn function which ta kes tha t w eight in to account. Giv en a complex W and comb inatorial map f : W → X , denote by k f k w, n the sum P σ ∈ W ( n ) w ( f ( σ )). F or a map f : α → X of a co m binato r ial n -sphere α to X , de no te the w eighted filling volume o f f by V ol w, n ( f ) = min k h k w, n +1 , wher e this minimum is taken ov er all maps h : β → X of co m binator ial ( n + 1)-balls to X with bo undary f . F or each n , the n th weigh ted geometric Dehn function of X , d w, n X : N → N , is defined via the form ula d w, n X ( k ) := max V ol w, n +1 ( f ) where the ma xim um is ta k en over all com binatorial maps f : α → X of n -spheres to X with k f k w, n ≤ k . I f the weigh t of each cell is set to o ne , then d n X and d w, n X are equal [Note: F or certain choices of w eig h ts, the geometr ic a nd weight ed geometric Dehn functions may be comparable. In general, howev er, if X has geometric Dehn functions and w eighted geometric Dehn functions defined in all dimensions, ther e need not be a n y par ticular r elation b et ween the tw o. If the weigh t function in each degree is a pr o per function on the set of simplicies, the weigh ted g eometric Dehn functions exist in a ll dimensions. In general, howev er, the weight ed geometric Dehn functions may fail to exist if the weigh t function fa ils to be prop er in one or more dimensions]. Assume a non-weigh ted weakly contractible complex X admits an action by a finitely generated gr o up G which is prop er and coc o mpact o n all finite skeleta. Then { d n X } n ≥ 0 are referred to as the geometric Dehn functions of G and denoted { d n G } n ≥ 0 . There is a natura l way to weigh t X so that the w eighted geo metric Dehn functions enco de infor mation ab out the gr oup action. Fix a basep oin t x 0 ∈ X (0) . F or a vertex v ∈ X (0) , set w X ( v ) := d X (1) ( x 0 , v ), the distance fro m v to the base point in the 1-skeleton of X , where each edge in the 1 -sk eleton is a ssumed to ha ve length 1. F or an n -cell σ ∈ X ( n ) with vertices ( v 0 , . . . , v k ), let w X ( σ ) := P k i =1 w X ( v i ), the sum of the w eig h ts of the v er tices. Changing ba sepoints yields differe n t w e ig h t functions, but if fo r each n there is a b ound on the num ber of vertices an n -cell can po ssess, then the t wo weight functions will b e linearly equiv alent. W e refer to this choice of assigning weigh ts to cells as the 1-skeleton weigh ting . F or this choice o f 12 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY weigh t on X there is, for e a c h n ≥ 0, a constant K n with w X ( g · σ ) ≤ K n · ℓ G ( g ) + w X ( σ ) for all n -cells σ (compare to ( 3 ) above), where ℓ G denotes the word-length function of G with resp e ct to so me fixed finite gener ating set. In this case, the G -action can be used to compare d n X with d w, n X . Up to equiv alence, one has [ 23 , Lemma s 2.3, 2 .4 ] (12) d w, n X ( k ) ≤ ( d n X ( k )) 2 , d n X ( k ) ≤ k d w, n X ( k 2 ) Given a finite sub complex b of X equipp ed with a 1-skeleton weight function w X , define the weigh t o f b to b e | b | w := P σ ∈ b w X ( σ ). 2.5.2. The algebr aic setting. In the literature, it is the g eometric Dehn functions that hav e received the mos t attention. Our fo cus, how ever, will be on the homo- logical version of the a bov e co ns tructions. Supp ose ( C ∗ , d ∗ ) is a n acyclic chain complex of fr ee Z -mo dules. Deno te by { v i | i ∈ I } , a basis of C n ov er Z . Giv en an element α = P i ∈I λ i v i of C n , set k α k n := P i ∈I | λ i | . F or α ∈ C n a cycle, let V ol n +1 ( α ) := min k β k n +1 , wher e this minimum is taken over all β ∈ C n +1 with d n +1 ( β ) = α . Define a function d n C : N → N b y d n C ( k ) := max { V ol n +1 ( α ) | d n ( α ) = 0 , k α k n ≤ k } The function d n C is the n t h un weigh ted homological Dehn function of C ∗ . These Dehn functions measure the filling complexit y of the chain complex C ∗ . As befor e, these Dehn functions do no t exist if the cor respo nding maximum v alues do not exist. When the C n come equipp ed with a weigh t function w on its basis, homological Dehn functions ca n be defined so as to tak e that weigh t in to account. F or α = P i ∈I λ i v i , let k α k w, n := P i ∈I | λ i | w ( v i ). If α is a c ycle, the w eighted filling volume of α is V ol w, n +1 ( α ) := min k β k w, n +1 , wher e this minimum is taken ov er all β ∈ C n +1 with d n +1 ( β ) = α . Define the n t h weigh ted Dehn function of C ∗ , d w, n C : N → N , by d w, n C ( k ) := max { V ol w, n +1 ( α ) | d n ( α ) = 0 , k α k w, n ≤ k } If the w eight of each basis element is se t to o ne, then d n C = d w, n C . F or certain choices of weight s, the unw eighted and the weigh ted Dehn functions ma y b e c omparable. In g eneral, howev er, no relations hip needs exist betw een the t w o. Now suppo se G is an F P ∞ group e q uipped with word-length function L , and C ∗ is a re solution o f Z over Z [ G ] which in ea c h deg ree is a finitely gener ated free mo dule ov er Z [ G ]. In this ca se, the collectio n { d n C } are referred to as the Dehn functions o f G , denoted d n G . W e will call the resolution C ∗ k -nice ( k ≤ ∞ ) if • for each finite n ≤ k , C n = Z [ G ][ T n ] for so me finite w eighted set ( T n , w T n ); • for each finite n ≥ 0 , S n (= the orbit o f T n under the free a ction of G ) is equipp e d with a prop er weigh t function w S n satisfying C 1 ,n L ( g ) + w T n ( t ) ≤ w S n ( g t ) ≤ C 2 ,n L ( g ) + w T n ( t ) ∀ g ∈ G, t ∈ T n for positive constant s C 1 ,n ≤ C 2 ,n depe nding only on n ; • for each finite n , d C n : C n → C n − 1 is linear ly bounded with resp ect to the weigh t functions on C n and C n − 1 . The term “nice ” will refer to the cas e k = ∞ . The weight ed Dehn functions of G (through dimensio n k if k is finite) are g iv en b y { d w, n G := d w, n C } nn ≥ 0 , implying B D ∗ is a c ontinuous re solution of C over H B ,L ( G ) thr ough dimension k . Pr o of. No te first that the “ niceness” of C ∗ guarantees that bo undary map d C n : C n → C n − 1 extends to a contin uous b oundary map d n : B D n → B D n − 1 . W e will prov e the lemma in three s teps. Claim 1 F or each k > n ≥ 1, there exists a function f n ∈ B ′ so that for all α ∈ ker( d C n ) and h ∈ B , there exists β α ∈ C n +1 with d C n ( β α ) = α , and | β α | h ≤ ( h ◦ f n ) ( k α k w, n ). Pr o of . The hypothesis o n { d w, n C } implies that for each n < k there exists f n ∈ B ′ with V ol w, n +1 ( α ) ≤ f n ( k α k w, n ) ∀ α ∈ k er( d C n ) Then for α = P m ij g i t j ∈ ker( d C n ), w e may c ho ose β α = P n kl g k s l ∈ C n +1 with d C n +1 ( β α ) = α and k β α k w, n +1 = X | n kl | w n +1 ( g k s l ) ≤ f n  X | m ij | w n ( g i t j )  = f n ( k α k w, n ) Since B  L , w e may assume that h ∈ B is sup er-additive on the interv al [1 , ∞ ). One then ha s | β α | h =    X n kl g k s l    h := X | n kl | h ( w n +1 ( g k s l )) ≤ h  X | n kl | w ( g k s l )  by the sup er-additivit y of h = h  k β α k w, n +1  ≤ ( h ◦ f n ) ( k α k w, n ) // Claim 2 F or each k > n ≥ − 1 there exists a B - bounded linear section s C n +1 : C n → C n +1 satisfying d C n +1 s C n +1 + s C n d C n = I d . Pr o of . The case n = − 1 is trivial since any basis elemen t of C 0 determines a linea r injection Z = C − 1 → C 0 which is b ounded. Assume s C n has b een defined. Let p n = ( I d − s C n d C n ) : C n → ker( d C n ); this pro jection onto k er( d C n ) is b ounded via the b oundedness of s n . Thus we may find an f ′ n ∈ B ′ with 14 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY k p n ( x ) k w, n ≤ f ′ n ( k x k w, n ) for all x ∈ C n . Let f ′′ n = f n ◦ f ′ n . F or each basis element g i t j ∈ C n , set s C n +1 ( g i t j ) = β p n ( g i t j ) , as defined in the a bov e Claim. Then for ea c h supe r -additiv e h ∈ B ,   s C n +1 ( g i t j )   h =   β p n ( g i t j )   h ≤ ( h ◦ f n ) ( k p n ( g i t j ) k w, n ) ≤ ( h ◦ f n ◦ f ′ n ) k g i t j k w, n = ( h ◦ f ′′ n )( w n ( g i t j )) Extending s C n +1 linearly to a ll of C n yields the desir ed result. // Claim 3 F or each k > n ≥ − 1, the linea r extension of s C n +1 to D n yields a B -b ounded linear map s D n +1 : D n → D n +1 . Pr o of . This follows from the sequence of inequalities    s D n +1  X λ ij g i t j     h = X | λ ij |   s C n +1 ( g i t j )   h ≤ X | λ ij | | g i t j | h ◦ f ′′ n =    X λ ij g i t j    h ◦ f ′′ n // This completes the pro of o f the lemma. W e should note that the definition of weigh ted Dehn function co uld be considered for more genera l r esolutions of C o ver C [ G ] whic h do not arise fr om tenso ring a nice resolution of Z ov er Z [ G ] with C . How ever, for this more genera l class, it is likely that the statement of this lemma no longer holds true.  Although it a ppears that not a llowing your prope r weight function w to take v alues in (0 , 1 ) is res tr ictiv e, it is a lw ays p ossible to find a linear ly equiv alent prop er weigh t function w ′ which takes v alues in Z + . While the weight structure ma y change slight ly , the co mpletio ns ar is ing from using the tw o weights will a gree. In many cases, for example the 1-skeleton w eighting discus s ed above and in [ 35 ], the naturally occurr ing weigh t function is integral-v alued. Also, in the pro of of Cla im 1 a bov e, we assumed the existence o f a function h ∈ B which was sup eradditiv e on [0 , 1). T o this end supp ose f and g are differentiable functions, and co ns ider the following prop erty: There exists a C ≥ 0 s uc h that for all x ≥ C f ( g ( x )) ≥ g ( f ( x )) . A sufficien t set of conditions to guar an tee this is: (1) f ( g ( C )) ≥ g ( f ( C )). (2) [ f g ( x ))] ′ ≥ [ g ( f ( x ))] ′ . Restrict to the case where g is the linear function g ( x ) = r x , for r ≥ 1 a real nu mber. The n ( f ( g ( x ))) ′ = ( f ( rx )) ′ = rf ′ ( rx ) ( g ( f ( x ))) ′ = rf ′ ( x ) In this case condition (2) is just the requirement that f ′ ( rx ) ≥ f ′ ( x ) , r ≥ 1, i.e., that f ′ is non-decreasing. If f (0) is requir ed to b e > 0, then taking, say , C = 1 this B -BOUNDED COHOMOLOGY AND APPLICA TIONS 15 condition b ecomes f ( r ) ≥ rf (1). If B  L , this s ho ws that B con tains functions which are supera dditiv e when r estricted to [1 , ∞ ). Corollary 1. L et G b e a fin itely gener ate d gr oup acting pr op erly and c o c omp actly on a c ontr actible p olyhe dr al c omplex X , with finit ely many orbits in e ach dimension, and endowe d with t he 1 -skeleton weighting. If the inte gr al p olyhe dr al chain c omplex, C n ( X ; Z ) , admits B ′ b ounde d weighte d Dehn fu n ctions and B is a right B ′ -class, then the F r e chet c ompletion B C n ( X ; C ) gives a c ontinuous r esolution of C over H B ,L ( G ) . In p articular if C n ( X ; Z ) admits p olynomial ly b ounde d weighte d Dehn functions, so do es C n ( X ; C ) . Remarks • It is a r e sult due to Gersten that for finitely-pr esen ted gr oups, the first algebraic and first g e o metric Dehn functions are equiv a len t. Ho wev er, in dimensions greater than o ne, it is not a t all clear if such a re lation p e rsists even when both types are defined. The one ca s e in which one ca n prov e an e quiv alence is when there is a G − H F ∞ mo del for E G admitting a n appropria te “coning” opera tion in all dimensions with explicitly computable bo unds on the num b er and weigh ts of the simplices used in coning off a simplex of one lower dimension (such is the case when G is a sync hronously combable - see below). • F o r finitely generated groups, w o rd-h ype rbolicity is e q uiv a len t to ha ving d 1 G bo unded by a linea r function. Hyperb olic g roups provide in ter esting phenomena in the co n text o f Dehn functions . A prime example is the isop erimetric gap. If d 1 G is b ounded by a function of the form n r with r < 2, then d 1 G is bo unded b y a linear function [ 18 , 36 ]. In particular if G is no t hyperb olic, then d 1 G m ust be at least quadratic. O n the other hand, it is well known that for a h yp erb olic g r oup G , the functions d n G are linearly b ounded in every dimension n . The geometric ch aracter ization of the iso cohomolog ic a l pr operty discusse d in Section 3.2 below implies that d w, n G are all linearly b ounded Dehn functions, pro viding a bounded v ersion of the F P α condition described ab o ve. This idea of combining boundednes s with the F P k condition is made precise b y Definition 1 . Given a b ounding class B and a gr oup with wor d-length ( G, L ) , we say G is of typ e B F P k ( k < ∞ ) if it is of typ e F P k , and ther e exists a k -nic e r esolution D ∗ of C over C [ G ] for which the c ompletion B D ∗ admits a b ounde d line ar chain c ontr action thr ough dimension k . We say G is B F P ∞ if it is B F P k for al l k . 2.6. Pro ducts, copro ducts and pairing op erations. Definition 2. L et ( X , w ) b e a weighte d set. A b ounding class B is nucle ar for ( X , w ) if for every λ ∈ B ther e is η ∈ B such t hat the fol lowing series c onver ges. X x ∈ X λ ( w ( x )) η ( w ( x )) F or example, if ( X , w ) has p olynomial gr o wth then P is a nuclear b ounding cla s s, but if ( X , w ) has exp onential growth P is not n uclear. The exp onential bounding class is n uclea r for every finitely generated g roup with word-length. The following lemma motiv a tes this definition. 16 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Lemma 2. L et ( G, L ) b e a gr oup with a pr op er length fun ction, and let B b e a nucle ar b ounding class for ( G, L ) . Then H B ,L ( G ) is a nucle ar F r e chet algebr a. F or a borno logical space V , let V ′ denote the dual space. Our interest in nucle- arity arises from its use in iden tifying  V ˆ ⊗ W  ′ . Lemma 3. L et V and W b e F r e chet sp ac es, and let W b e nucle ar. Then  V ˆ ⊗ W  ′ = V ′ ˆ ⊗ W ′ . Pr o of. B y definition,  V ˆ ⊗ W  ′ = Hom bdd ( V ˆ ⊗ W, C ). Using the adjointness of the pro jectiv e tensor pro duct, this is iso morphic to Hom bdd ( V , Hom bdd ( W , C )) = Hom bdd ( V , W ′ ). As W is a nuclear F rechet algebr a, its dua l is also n uclear . Corol- lary 1.161 of [ 28 ] g iv es that Hom bdd ( V , W ′ ) ∼ = V ′ ˆ ⊗ W ′ .  This Lemma suggests tha t an y so rt of K ¨ unneth Theorem in B -bounded coho- mology would ho ld o nly under very res tr ictiv e conditions. Nevertheless, the pairing op erations used to prov e it exist in the B -b ounded setting under minimal co nditions. W e consider them next, as they will b e needed later on. Let ( X , w X ) and ( Y , w Y ) b e a weigh ted simplicial sets. Then their pro duct ( X × Y , w X × w Y ) is again a w eighted simplicia l set, wher e X × Y is equipped with diagonal simplicial structure. It follows from the definition of a weigh ted complex that the Alexander- Whitney map ∆ AS : C ∗ ( X × Y ) → C ∗ ( X ) ⊗ C ∗ ( Y ) is uniformly bo unded ab o ve in each dimensio n n b y a linea r function of n . As a result, when B is multiplicativ e this induces an exterio r algebra ic tensor pro duct on the co chain level B ∆ ∗ : → B C ∗ ( X ) ⊗ B C ∗ ( Y ) → B C ∗ ( X × Y ) F rom the definitions o f H ∗ ( − ) and H ∗ ( − ) there is an obvious Kro nec ker-Delta pairing B H ∗ ( X ) ⊗ B H ∗ ( X ) → C ( c, d ) 7→ < c, d > , x = a, b More gener ally , an analysis of the standar d ca p pro duct op e ration on the chain and co c ha in level yields a cap pro duct oper ation B H ∗ ( X ) ⊗ B H ∗ ( X ) → B H ∗ ( X ) , ( c, d ) 7→ c ∩ d, x = a, b These op erations satisfy the appropriate comm uting diagrams with resp ect to the comparison map Φ ∗ B and Φ B ∗ , lea ding to the iden tities < Φ ∗ B ( c ) , d > = < c, Φ B ∗ ( d ) >, c ∈ B H ∗ ( X ) , d ∈ H ∗ ( X ) (13) Φ ∗ B ( c ) ∩ d = c ∩ Φ B ∗ ( d ) , c ∈ B H ∗ ( X ) , d ∈ H ∗ ( X ) (14) 3. B cohomology of discrete groups 3.1. Comba ble groups. Call a function σ : N → N a reparameteriza tion if • σ (0) = 0, • σ ( n + 1) equals either σ ( n ) or σ ( n ) + 1, • lim n →∞ σ ( n ) = ∞ B -BOUNDED COHOMOLOGY AND APPLICA TIONS 17 Definition 3 . L et ( X , ∗ ) b e a discr ete met ric sp ac e with b asep oint. By a c ombing of X we wil l me an a c ol le ction of functions { f n : X → X } n ≥ 0 satisfying (C1) f 0 ( x ) = x ∀ x ∈ X (C2) Ther e exists a sup er-additive function ψ such that ∀ x, y ∈ X , ther e ar e r ep ar ameterizations σ and σ ′ with d ( f σ ( n ) ( x ) , f σ ′ ( n ) ( y )) ≤ ψ ( d ( x, y )) for al l n ≥ 0 . (C3) ∃ λ such that ∀ x ∈ X , n ∈ N , d ( f n ( x ) , f n +1 ( x )) ≤ λ (C4) ∃ φ such that f n ( x ) = ∗ ∀ n ≥ φ ( d ( x, ∗ )) Remarks: • As noted in the intro ductio n, the combings above are oriented in the op- po site dir ection than what has b een customarily the c a se. • Axiom (C2) allows for what are typically refer red to as a sync hr onous com b- ings, with synchronous combings corresp onding to the c ase that the r epa- rameteriza tio ns are the identit y ma ps . Note also that • the reparameter izations σ , σ ′ in (C2) dep end on x and y . Definition 4. Given a discr ete gr oup G e quipp e d with a (pr op er) length fun ction L , a c ombing of G (with r esp e ct to L ), or ( G, L ) , is a c ombing of the discr ete metric sp ac e ( G, d L ) , wher e d L ( g 1 , g 2 ) := L ( g − 1 1 g 2 ) . W e first show that repar ameterizations can b e chosen so as to b e co mpatible on sp ecific ( n + 1)-tuples. Lemma 4. Supp ose ( G, L ) admits an asynchr onous c ombing in the ab ove sens e. Then for al l ( m + 1 ) - tuples ( g 0 , . . . , g m ) ∈ G ( m +1) , ther e exist r ep ar ameterizations σ 0 , . . . , σ m such that ∀ n ≥ 0 , d ( f σ i ( n ) ( g i ) , f σ i +1 ( n ) ( g i +1 )) ≤ ψ ( d ( g i , g i +1 )) Pr o of. B y definition it is true for m = 1. Assume then it is true for fixed m ≥ 1 . Given an ( m + 2)-tuple ( g 0 , . . . , g m +1 ), w e may assume b y inductio n that • There exist reparameterizatio ns σ 0 , . . . , σ m with d ( f σ i ( n ) ( g i ) , f σ i +1 ( n ) ( g i +1 )) ≤ ψ ( d ( g i , g i +1 ) for all n ≥ 1, and • There exist reparameterizatio ns σ ′ m , σ ′ m +1 with d ( f σ ′ m ( n ) ( g m ) , f σ ′ m +1 ( n ) ( g m +1 )) ≤ ψ ( d ( g m , g m +1 )) for all n ≥ 1 W e need to show that the repar ameterization functions can be further reparam- eterized so as to sync hronize σ m and σ ′ m . F or this we pro ceed by induction on k ∈ N = the domain of the re pa rameterization functions. k = 0 By definition, σ m (0) = σ ′ m (0) = 0. k > 0 Supp ose σ m ( i ) = σ ′ m ( i ) for 0 ≤ i ≤ k . Case 1 σ m ( k + 1) = σ ′ m ( k + 1). In this cas e there is nothing to do. Case 2 σ m ( k + 1) = σ m ( k ) , σ ′ m ( k + 1) = σ ′ m ( k ) + 1. In this case we leav e σ m alone, a nd r e de fine σ ′ m , σ ′ m +1 : for l = m, m + 1 , ( σ ′ l ) new ( i ) = ( ( σ ′ l ) old ( i ) 0 ≤ i ≤ k ( σ ′ l ) old ( i ) − 1 k + 1 ≤ i 18 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Case 3 σ m ( k + 1) = σ m ( k ) + 1 , σ ′ m ( k + 1) = σ ′ m ( k ). In this case we leave σ ′ m alone, a nd r e de fine σ 0 , . . . , σ m : for 0 ≤ l ≤ m , ( σ l ) new ( i ) = ( ( σ l ) old ( i ) 0 ≤ i ≤ k ( σ l ) old ( i ) − 1 k + 1 ≤ i Thu s b y induction on k , we may choose repara meterization functions σ 0 , . . . , σ m , σ ′ m , σ ′ m +1 with (S1) σ 0 , . . . , σ m satisfying the conditions of the L emma for the ( m + 1)-tuple ( g 0 , . . . , g m ), (S2) σ ′ m , σ ′ m +1 satisfying the conditions of the Lemma fo r the pair ( g m , g m +1 ), (S3) σ m = σ ′ m Setting σ m +1 := σ ′ m +1 then concludes the pro of of the initial induction step, and hence of the Lemma.  Definition 5. We say that a metric sp ac e ( X , d ) is quasi-ge o desic if ther e exist p ositive c onstants ǫ , S , and C such that for any two p oints x , y ∈ X , ther e is a finite se quenc e of p oints x 0 = x, x 1 , x 2 , . . . , x k = y satisfyi ng: (1) ǫ ≤ d ( x i , x i +1 ) ≤ S for i = 0 , 1 , . . . , k − 1 . (2) d ( x 0 , x 1 ) + d ( x 1 , x 2 ) + . . . + d ( x k − 1 , x k ) ≤ C d ( x, y ) . As w e are concerned primarily with co nnected complexes , all metric space s we consider will b e assumed to b e q ua si-geo desic. Lemma 5. Supp ose { f n : X → X } n ≥ 0 is an asynchr onous c ombing of a quasi- ge o desic metric sp ac e ( X , d ) . Ther e exists a p ositive c onstant K such t ha t for al l x, y ∈ X , ther e ar e r ep ar ameterizations σ and σ ′ such that for al l n ≥ 0 , d ( f σ ( n ) ( x ) , f σ ′ ( n ) ( y )) ≤ K d ( x, y ) . Pr o of. Le t x 0 = x, x 1 , x 2 , . . . , x k = y b e given by the quasi-g eodesic pro perty . By lemma 4 there are r eparameterizations, σ i , such that d ( f σ i ( n ) ( x i ) , f σ i +1 ( n ) ( x i +1 )) ≤ ψ ( S ), for all n. As k ≤ C ǫ d ( x, y ), the triang le inequality yields d ( f σ 0 ( n ) ( x ) , f σ k ( n ) ( y )) ≤ ψ ( S ) C ǫ d ( x, y ), for all n.  The next theorem was originally shown fo r synchronously combable gr oups in [ 1 ], and asynchronously co m bable gro ups thr ough dimension 3 in [ 1 7 ]. Our method of pr oof actually prov es more, as we will see in the following section. Theorem 1. If ( G, L ) admits an asynchr onous c ombing in the ab ove sense, then it is typ e H F ∞ . Pr o of. Le t E G . deno te the simplicial homogeneo us bar res o lution o f G . Le t G act in the usual wa y on the left, by g · ( g 0 , g 1 , . . . , g n ) := ( gg 0 , g g 1 , . . . , g g n ). Define a G -inv ar ian t simplicial weight function on E G . by w n ( g 0 , g 1 , . . . , g n ) := n − 1 X i =0 d ( g i , g i +1 ) = n − 1 X i =0 L ( g − 1 i g i +1 ) Because L is pro per, the orbit { ( g 0 , g 1 , . . . , g n ) | w n ( g 0 , g 1 , . . . , g n ) ≤ N } /G is a finite set for each n and N . This orbit may a lternativ ely b e describ ed as π − 1 ( B N ( B G n )), where B G . is the non-homogeneous bar construction o n G , π : E G . → B G . is given by π ( g 0 , g 1 , . . . , g n ) = [ g − 1 0 g 1 , g − 1 1 g 2 , . . . , g − 1 n − 1 g n ], and B N ( − ) denotes the N - ba ll B N ( B G n ) := { [ g 1 , . . . , g n ] | P n i =1 L ( g i ) ≤ N } . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 19 Recall that giv en two simplicial functions h 0 , h 1 : E G → E G , there is a ho motop y betw een them repr esen ted b y the “sum” H ( h 0 , h 1 )( g 0 , g 1 , . . . , g n ) = n X i =0 ( − 1) i ( h 0 ( g 0 ) , h 0 ( g 1 ) , . . . , h 0 ( g i ) , h 1 ( g i ) , h 1 ( g i +1 ) , . . . , h 1 ( g n )) Although we ha ve written this sum algebra ically , this should be viewed as a geometric sum which as sociates to the n -simplex ( g 0 , g 1 , . . . , g n ) the co llection of ( n + 1)-simplices indicated by the r igh t-ha nd side, with orientation determined by the co efficient ( − 1) i . Geometrically , this collection of ( n + 1) simplices, all of dimension ( n + 1), fit together to for m a s ubs e t whose geometr ic realization is home- omorphic to ∆ n × [0 , 1]. In fact, this last s tatemen t is true for more general types of maps which are not simplicial. In particular, given i) a fix e d asynchronous co m bing { f n } of G , ii) a fix e d n -simplex ( g 0 , . . . , g n ) o f E G . , and iii) a collectio n o f repara m- eterizations σ 0 , . . . , σ n satisfying the condition o f Lemma 1 with resp ect to i) and ii), w e ma y consider the ‘homo top y” fr o m ( f σ 0 ( m ) ( g 0 ) , f σ 1 ( m ) ( g 1 ) , . . . , f σ n ( m ) ( g n )) to ( f σ 0 ( m +1) ( g 0 ) , f σ 1 ( m +1) ( g 1 ) , . . . , f σ n ( m +1) ( g n )) giv en by the express ion (15) H σ ( { f k } ; m, m + 1)( g 0 , g 1 , . . . , g n ) := n X i =0 ( − 1) i ( f σ 0 ( m ) ( g 0 ) , f σ 1 ( m ) ( g 1 ) , . . . , f σ i ( m ) ( g i ) , f σ i ( m +1) ( g i ) , f σ i ( m +1) ( g i +1 ) , . . . , f σ i ( m +1) ( g n )) This is not par t of a globa l homotopy , but still yields a collection o f oriented ( n + 1)-simplices whose realization is homeomorphic to ∆ n × [0 , 1 ]. Moreover, these ho - motopies may b e str ung together, as the “end” of H σ ( { f k } ; m, m + 1)( g 0 , g 1 , . . . , g n ) and the “b eginning” of H σ ( { f k } ; m + 1 , m + 2)( g 0 , g 1 , . . . , g n ) matc h up. Given a function f : R + → R + , write w f n for the weight function w f n ( g 0 , g 1 , . . . , g n ) := n − 1 X i =0 f ( d ( g i , g i +1 )) By Lemma 1 and prope r t y (C3), w n +1 ( H σ ( { f k } ; m, m + 1)( g 0 , g 1 , . . . , g n )) (16) < ( n + 1 )  w n ( f σ 0 ( m ) ( g 0 ) , f σ 1 ( m ) ( g 1 ) , . . . , f σ n ( m ) ( g n )) + w n ( f σ 0 ( m +1) ( g 0 ) , f σ 1 ( m +1) ( g 1 ) , . . . , f σ n ( m +1) ( g n )) + λ  ≤ (2 n + 2)  w ψ n ( g 0 , g 1 , . . . , g n ) + λ  ≤ (2 n + 2) ψ ( w n ( g 0 , g 1 , . . . , g n )) + (2 n + 2) λ Equation ( 16 ) implies that every simplex in π − 1 ( B N ( B G n )) can be coned o ff in π − 1 ( B N ′ ( B G n )) where N ′ = (2 n + 2)( ψ ( N ) + λ )). Of cour s e, degenera cies preser v e the inequality . In other words, if ( g 0 , g 1 , . . . , g n ) = s I ( g ′ 0 , . . . , g ′ k ) for some iterated degeneracy map s I and k -simplex ( g ′ 0 , . . . , g ′ k ), then the inequality in ( 16 ) may be improv ed to w n +1 ( H σ ( { f k } ; m, m + 1)( s I ( g ′ 0 , . . . , g ′ k )) < (2 k + 2) ψ ( w k ( g ′ 0 , . . . , g ′ k )) + (2 k + 2) λ 20 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Let X ( n ) := E G . ( n ) , the simplicial n - sk ele ton of E G . F or each int eger N , le t X ( n ) N := X ( n ) ∩ π − 1 ( B N ( B G . )). Then X ( n ) is a n n -go o d complex for G in the sense of [ 10 ], and obviously X ( n ) = lim N X ( n ) N . Moreover, equations ( 15 ) and ( 16 ) together imply (17) X ( n ) N ֒ → X ( n ) N ′ is n ull-homo to pic , N ′ = (2 n + 2)( ψ ( N ) + λ ) By Theor em 2.2 of [ 10 ], we co nclude that G is of t yp e F P n . Then, as G is of t yp e F P n for eac h n , it m ust b e of t yp e F P ∞ [ 11 ].  In fact, the explicit es timates in ( 16 ) and ( 17 ) allow o ne to conclude a bit more. W e will ne e d some terminology . Definition 6. A discr ete gr oup with wor d-length ( G, L ) is B -c omb able (i.e., B - asynchr onously c omb able) if the functions ψ and φ in (C2)and (C4) ar e b ounde d ab ove by functions in the b ounding class B . As indicated a b ov e, g iven a bo rnological H B ,L ( G )-module V , one has the sub- co c ha in co mplex B C ∗ ( G ; V ) ⊂ C ∗ ( G ; V ) = Hom G ( C ∗ ( E G . ) , V ) consisting of those co c ha ins which are bo unded in the b ornology induced by B . The gro up G , or pair ( G, L ) is ca lle d B -isoco homological with resp ect to V (a bbr . V - B IC)) if the inclu- sion B C ∗ ( G ; V ) ⊂ C ∗ ( G ; V ) induces a n isomor phism of cohomo logy gr oups in all degrees. B H ∗ ( G ; V ) := H ∗ ( B C ∗ ( G ; V )) ∼ = → H ∗ ( G ; V ) B -iso cohomologica lit y with respect the trivial mo dule C is referred to simply as B -iso cohomologica l ( B -IC). The pair ( G, L ) is stro ngly B -iso cohomolog ical (abbr. B -SIC) if it is V - B IC for all b ornologica l H B ,L ( G )-modules V . Corollary 2. L et G b e a finitely-pr esent e d gr oup e quipp e d with wor d-length funct io n L , and B a multiplic ative b ounding class. If ( G, L ) is asynchr onously B -c omb able, then G is B -IC. Pr o of. B y the previous theorem, the h ypo thesis that G is B -as ync hr o nously co m- bable implies by equations ( 16 ) and ( 17 ) that in using the combing to co ne off a simplex of w e igh t m , b oth the n um ber of simplices appe a ring in the cone, a s well as the weigh t of each, is b ounded ab ov e by f i ( m ) where f i ∈ B . By the multiplicativit y of the b ounding c la ss B , the b ornologica l chain complex B C ∗ ( E G ) is a temp ered complex in the sense of Meyer [ 25 ] which satisfies the necessar y conditions estab- lished b y Meyer to co nclude the result (Meyer’s original r esult was stated only fo r the p olynomial, subexp onential and simple exp onential b ounding c la sses, but the same argument works for ar bitrary multiplicativ e b ounding classes).  3.2. B -iso cohomo l ogicalit y and t yp e B F P ∞ groups. It is natural to ask a b out the r elation b et ween the purely homolog ical no tion of strong B -iso cohomologicality and the mor e geometric/to polog ical B F P ∞ condition. The following result answers that que s tion; it is a gener alization to arbitr ary bo unding cla sses of [ 23 , Thm. 2.6 ]. Theorem 2. Le t G b e a fi nitely pr esente d gr oup of typ e F P r , for some r ≤ ∞ . F or k < r , the fol lowing ar e e quivalent. ( B 1) B H ∗ ( G ; V ) → H ∗ ( G ; V ) is an isomorp hism for al l b ornolo gic al H B ,L ( G ) - mo dules V , in al l de gr e es ∗ ≤ k + 1 . ( B 2) B H ∗ ( G ; V ) → H ∗ ( G ; V ) is surje ct ive for al l b ornolo gic al H B ,L ( G ) -mo dules V , in al l de gr e es ∗ ≤ k + 1 . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 21 ( B 3) G is B F P k . Before pro ceeding to the pro of of this theorem, we in tro duce necessar y notatio n and isolate tw o lemmas that highlight the importance of the finiteness assumption on G . F or a simplicial co mplex X equipp ed with the 1-skeleton weigh t function w X , and a free G ac tio n, co ns ider B B m ( X ) = ∂ m +1 ( B C m +1 ( X )) ⊂ B C m ( X ). On B B m ( X ) one has filling seminorms k k f ,λ , λ ∈ B , defined a s follows: k b k f ,λ = inf {k a k λ | ∃ a ∈ C m +1 ( X ) s.t. b = ∂ ( a ) } . This norm identifies B B m ( X ) with B C m +1 ( X ) / k er m +1 . Given a borno lo gical H B ,L ( G )-module V with b ornology defined by a collection of seminor ms 4 { η i } i ∈ I , an m -co ch ain c ∈ C m ( X ; V ) is B -b ounded (i.e., lie s in the subspace B C m ( X ; V )) if ∀ η i , i ∈ I ∃ λ ′ ∈ B s.t. ∀ σ ∈ X m , η i ( c ( σ )) ≤ λ ′ ( w X ( σ )) = k σ k λ ′ . Then, as was shown in [ 21 ], B H ∗ G ( X ; V ) := Ext ∗ H B ,L ( G ) ( C , V ) = the cohomolog y of the co c ha in complex { B C m ( X ; V )) } m ≥ 0 . In particular, B B m ( X ), e q uipped with the collec tio n of filling seminor ms defined a bov e, is a b ornologica l mo dule over H B ,L ( G ), and so an m -chain c ∈ B C m ( X ; B B m ( X )) sa tisfies ∀ λ ∈ B ∃ λ ′ ∈ B s.t. ∀ σ ∈ X m , k c ( σ ) k f ,λ ≤ λ ′ ( w X ( σ )) = k σ k λ ′ . F or a weigh ted set ( U, w U ), we can (as in [JOR1]) form the weigh ted vector space X ( U ) := C [ U ], with weigh ting given on basis elements by w U . F o r any b ounding class B and λ ∈ B , we hav e an as socia ted seminorm k P i α i x i k λ := P i | α i | λ ( w ( x i )). If ( V , w V ) is a second weigh ted s et, a linear map φ : X ( U ) → X ( V ) is B -b ounded if for all λ ∈ B , ∃ λ ′ ∈ B such that k φ ( x ) k λ ≤ k x k λ ′ ∀ x ∈ X ( U ). Let S a nd T be fre e G -s ets equipp ed with weigh t functions w S and w T , s uch that S/G is finite. Let { s 1 , s 2 , . . . , s N } ⊂ S b e a complete set of or bit r e pr esen tatives satisfying w S ( s j ) ≤ w S ( g s j ) for all 1 ≤ j ≤ N , g ∈ G . W e a ssume the existence of constants D 1 , D 2 , D 3 satisfying (1) w ( gz ) ≤ D 1 L ( g ) + w ( z ) ∀ z ∈ Z = S, T , w = w S , w T (2) L ( g ) ≤ D 2 w S ( g s j ) + D 3 ∀ 1 ≤ j ≤ N , g ∈ G Lemma 6. L et S, T b e as ab ove, and V , r esp e ctively W , denote the weighte d ve ctor sp ac e over C with b asis S , r esp e ctively T . Then any G -e quivariant line ar map V → W is B -b ounde d. Let W ′ ⊂ B W b e a subspace clos ed in the F rechet b ornology (= top ology), and set W ′′ = B W/ W ′ . Suppo se we a re also giv en a G -equiv aria n t linear map h : V → W ′′ . Then h can be lifted to a G -equiv ariant linea r map f : V → B W . By the pre vious lemma, this lifting is B -b ounded. Hence Lemma 7. Any G -e quivariant line ar map V → W ′′ is B -b oun de d. Pr o of of The or em 2 . ( B 1) obviously implies ( B 2 ). The implication ( B 3 ) ⇒ ( B 1) follows by a natural extension o f the arguments of [ 23 ] and [ 3 4 ]. Namely , the B F P k condition yields a complex B D ∗ , which is the completion of a k -nice resolution of C ov er C [ G ], a dmitting a b ounded linea r c hain co n tra ction through dimensio n k . F or 4 W e remark that this is an assumption on V . Not ev ery b ornological space is of this form. 22 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY degrees less than or eq ua l to k , this complex is a free H B ,L ( G )-module o n finitely many generato rs. This implies the result. The main p oin t is to show ( B 2) implies ( B 3), sp ecifically that there exists a resolution C ∗ of C over C [ G ] whic h is finite dimens io nal in degrees less than k , and who se completion B C ∗ with respe c t to the semi-nor ms induced by B yields a bo unded r esolution of C over H B ,L ( G ) through the appr opriate r a nge. This verification will be carried out in t wo pa rts. W e first use the epicohomologica l condition ( B 2) to show that for every λ ∈ B there is a λ ′ ∈ B such that for all b ∈ B m ( X ), k b k f ,λ ≤ k b k λ ′ . W e then show inductiv ely that this inequality guarantees the existence of a B -b ounded section ˜ s m +1 : C m → C m +1 . The B -b oundedness of the section implies that it extends to the B -completions as a b ounded linear splitting on B C ∗ . That G is finitely pr esen ted and F P r implies the existence of a contractible free G -simplicial complex X such that in dimensio ns n ≤ r the G action o n X ( n ) is cofinite. Fix a basepoint x 0 ∈ X (0) , and equip X ( r ) with the 1-skeleton w eighting. Also fix a family of r epresent atives o f the orbits of G in X (0) , R = { x 0 , x 1 , . . . , x l } with each x i satisfying d X (1) ( x i , x 0 ) ≤ d X (1) ( g x i , x 0 ) for all g ∈ G . Let I G = ⊔ l i =0 G i , where G i is a copy o f G . W e denote an elemen t of I G by ( g , i ) for g ∈ G and 0 ≤ i ≤ l . Let Y b e the geo metric realization of the homo g eneous ba r resolution on I G . Thus, C ∗ ( X ) and C ∗ ( Y ) are resolutio ns o f C ov er C [ G ]. W e endo w Y with a w eight function arising from the length function L on the group G , given on simplices b y w Y ([( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )]) := n X i =0 L ( g i ) . The weight function w Y satisfies the inequalit y w Y ( g σ ) ≤ ( n + 1) L ( g ) + w Y ( σ ) ∀ σ ∈ Y ( n ) while for w X there is a constant C related to the quas i-isometric equiv alence b e- t ween G and X ( r ) with w ( g σ ) ≤ C ( n + 1) L ( g ) + w ( σ ) ∀ σ ∈ X ( n ) . As in [ 23 , Th. 2.6], there exist C [ G ]-mo dule mor phis ms of chain complexes φ ∗ : C ∗ ( Y ) → C ∗ ( X ) and ψ ∗ : C ∗ ( X ) → C ∗ ( Y ) with b oth φ ∗ ◦ ψ ∗ and ψ ∗ ◦ φ ∗ G -equiv ariantly chain-homotopic to the identit y . Let V k − 1 = B B k − 1 ( X ) equipped with the filling seminorms, k k f ,λ , λ ∈ B , and let u ∈ C k G ( X ; V k − 1 ) be the k -cocy cle given as the comp osition C k ( X ) ∂ k → B k − 1 ( X ) ι k − 1 ֒ → V k − 1 . The pr operties o f φ and ψ ensure there is a G -equiv ariant ( k − 1 )-cocy c le v ∈ C k − 1 G ( X ; V k − 1 ) with u = ( ψ k ◦ φ k )( u ) + δ ( v ). F urthermore, condition ( B 2) guar- antees there is a G -equiv ariant B -bounded c o cycle u ′ ∈ B C k G ( Y ; V k − 1 ) and a G - equiv ariant ( k − 1)-co chain v ′ ∈ C k − 1 G ( Y ; V k − 1 ) satisfying the equation φ k ( u ) = u ′ + δ ( v ′ ). By the argument of [ 30 ], one has for any b ∈ B k − 1 ( X ) (18) ι k − 1 b = u ′ ([ e, ψ k − 1 ( b )]) + ( ψ k − 1 ( v ′ ) + v )( b ) where [ e, P γ [ g 0 ,...,g n ] [ g 0 , . . . , g n ]] := P γ [ g 0 ,...,g n ] [ e, g 0 , . . . , g n ]. By ( 18 ), one has k b k f ,λ ≤ k u ′ ([ e, ψ k − 1 ( b )]) k f ,λ +   ( ψ k − 1 ( v ′ ) + v )( b )   f ,λ . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 23 As u ′ is B - bounded, there exis ts a n element λ ′ ∈ B satisfying k u ′ ([ e, ψ k − 1 ( b )] k f ,λ ≤ k [ e, ψ k − 1 ( b )] k λ ′ = k ψ k − 1 ( b ) k λ ′ . T aking ( S, w S ) = ( X ( k − 1) , w X ) and ( T , w T ) = ( Y ( k − 1) , w Y ) and a pplying Lemma 6 shows tha t ψ k − 1 is B -b ounded. By Lemma 7 , the map ( ψ ( k − 1) ( v ′ ) + v ) : C k − 1 ( X ) → V k − 1 is als o b ounded. Hence the sum mu st b e b ounded. Thus for all λ ∈ B there is λ ′′ ∈ B such that for a ll b ∈ B B k − 1 ( X ) k b k f ,λ ≤ k b k λ ′′ . As B is w eakly count able, B C k ( X ) is a F r ec he t space. W e define a m ulti-v alue d map F : B B k − 1 ( X ) → B C k ( X ) b y asso ciating to an element b + ker ∂ k ∈ B B k − 1 ( X ), the affine subspace { a | ∂ k ( a ) = b } . This is a trans la te of the subspace ker ∂ k in B C k ( X ). W e fo llo w Noskov in [ 32 ] b y using a version of Michael’s Theor em, g e ner- alized to F r e c het spaces. By [ 13 , Thm. 3.4], F has a (no t necessarily linear) single- v alued contin uous selection. Th us, for ev ery b ∈ B B k − 1 ( X ) there is an a ∈ B C k ( X ) with ∂ k ( a ) = b such that for ev er y λ ∈ B , there is a λ ′ ∈ B satisfying k a k λ ≤ k b k f ,λ ′ . The last t wo inequalities imply that we hav e a co n tinuous map ∂ k B C k ( X ) → B C k ( X ), with ∂ k B C k ( X ) ⊂ B C k − 1 ( X ) top ologized by the countable family of norms, ( k · k λ ) λ ∈B . W e use these filling inequalities b elow to construct a b ounded contracting homotopy for B C ∗ ( X ), by modifying the methods of [ 30 ]. W e set C − 1 ( X ) = C , mak ing C ∗ ( X ) an aug men ted r esolution of C − 1 ( X ) ov er C [ G ]. Let ∂ be the bo undary map ∂ i : C i ( X ) → C i − 1 ( X ). (W e also call the bo undary map from B C i ( X ) to B C i − 1 ( X ) by ∂ i .) Let B C ∗ ( X ) b e the B -completio n of C ∗ ( X ). W e will show that B C ∗ ( X ) a dmits a b ounded linear contraction { S n : B C n ( X ) → B C n +1 ( X ) } through dimension k . F or a v ∈ X (0) , let i ( v ) b e such that v ∈ G · x i ( v ) . The aug mented complex B C ∗ ( I G ) is a free b ornologica l B -r e solution of C = B C − 1 ( I G ), and c omes equipp ed with a b ounded C - linear c on tr a ction. Lab el it s i : B C i ( I G ) → B C i +1 ( I G ), w ith the boundar y map d i : B C i ( I G ) → B C i − 1 ( I G ). Define G -equiv aria n t B -b ounded chain maps Ψ : B C ∗ ( X ) → B C ∗ ( I G ) and Φ : B C ∗ ( I G ) → B C ∗ ( X ) as follows. In dimensions n ≤ − 1 , set Ψ n : B C n ( X ) → B C n ( I G ) equa l to the identit y . (In degr ee − 1, this is just the map from C → C , and for n < − 1, B C n ( X ) a nd B C n ( I G ) a re b oth zero.) Le t Ψ 0 : B C 0 ( X ) → B C 0 ( I G ) b e the map given on simplices by [ v ] 7→ [( g , i ( v ))]. This is a well-defined, G -equiv aria n t linear map. More genera lly , for 1 ≤ n ≤ k let Ψ n : B C n ( X ) → B C n ( I G ) b e defined by [ v 0 , v 1 , . . . , v n ] 7→ [( g 0 , i ( v 0 )) , ( g 1 , i ( v 1 )) , . . . , ( g n , i ( v n ))]. W e note that these maps are B -b ounded by Lemma 6 . In order to cons truct the contin uous G -equiv ariant chain map Φ : B C ∗ ( I G ) → B C ∗ ( X ), w e first o bserv e that for each p oint v ∈ X (0) , there is a unique g v ∈ G and x i ∈ R such that g v x i = v , with the assignment v 7→ g v being G -equiv ar ian t. As G is quasi- isometric to X (1) , there exist p ositive constants A and B s uch that for all g ∈ G , and x j ∈ R , d X ( g x j , x 0 ) ≤ AL ( g ) + B . Define the G -eq uiv a rian t linea r map Φ 0 : C 0 ( I G ) → B C 0 ( X ) b y Φ 0 ([( g , i )]) = [ g x i ] . 24 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY F or λ in the b ounding clas s B , there is an λ ′ ∈ B such that λ ( Ax + B ) ≤ λ ′ ( x ). Then k Φ 0 ([( g , i )]) k λ ≤ k [( g , i )] k λ ′ . Therefore Φ 0 extends to a bounded B G -mo dule map Φ 0 : B C 0 ( I G ) → B C 0 ( X ). As Φ 0 ( d 1 [( g 0 , i 0 ) , ( g 1 , i 1 )]) = [ g 1 x i 1 ] − [ g 0 x i 0 ], this is a 0-b oundary in B 0 ( X ). By the filling inequalities ab o ve, there is a 1-chain c = c (( g 0 , i 0 ) , ( g 1 , i 1 )) ∈ B C 1 ( X ) such that fo r each λ ∈ B there is λ ′ ∈ B with k c (( g 0 , i 0 ) , ( g 1 , i 1 )) k λ ≤ k [( g 0 , i 0 ) , ( g 1 , i 1 )] k λ ′ . There co uld b e man y p ossible choices for c (( g 0 , i 0 ) , ( g 1 , i 1 )). W e make the fo llo wing assumptions on o ur choice: (1) If { g 0 x i 0 , g 1 x i 1 } spans a 1-simplex [ g 0 x i 0 , g 1 x i 1 ] in X (2) , then c (( g 0 , i 0 ) , ( g 1 , i 1 )) = [ g 0 x i 0 , g 1 x i 1 ]. (2) c is G -equiv ariant (as the action of G on X (2) is fr e e, this can alwa ys b e arrang ed). Set Φ 1 ([( g 0 , i 0 ) , ( g 1 , i 1 )]) := c (( g 0 , i 0 ) , ( g 1 , i 1 )). This yields a G -eq uiv a rian t B - bo unded linear map Φ 1 : B C 1 ( I G ) → B C 1 ( X ), satisfying ∂ 1 Φ 1 = Φ 0 d 1 . No te that we may assume (1) o nly b ecause X is equipped with the 1- sk e leton weigh t- ing; thus the w e ig h t o f a simplex is b o unded by the s um o f the weigh ts of the co dimension 1 faces. Inductively , s uppose that we ha ve constructed Φ n for some n ≤ k − 1, the ma p having b een defined thr o ugh the use of fillings c (( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )) of n- bo undaries Φ n − 1 ( d n ([( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )])) in X ( n − 1) satisfying: ( P 1 ,n ) If g 0 x i 0 , g 1 x i 1 , . . . , g n x i n span a n n-simplex [ g 0 x i 0 , g 1 x i 1 , . . . , g n x i n ] in X ( n ) , then c (( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )) = [ g 0 x i 0 , g 1 x i 1 , . . . , g n x i n ]. ( P 2 ,n ) The c hoice of c is G -equiv ariant. ( P 3 ,n ) F or every λ ∈ B there is λ ′ ∈ B such that k c (( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )) k λ ≤ k [( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )] k λ ′ for all [( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n , i n )] ∈ C n ( I G ). Consider the ( n + 1)-ch ain [( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n +1 , i n +1 )] ∈ C n +1 ( I G ). d n +1 ([( g 0 , i 0 ) , ( g 1 , i 1 ) , . . . , ( g n +1 , i n +1 )]) = n +1 X j =0 ( − 1) j [( g 0 , i 0 ) , . . . , \ ( g j , i j ) , . . . , ( g n +1 , i n +1 )] . Then Φ n ( d n +1 ([( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )])) = n +1 X j =0 ( − 1) j c (( g 0 , i 0 ) , . . . , \ ( g j , i j ) , . . . , ( g n +1 , i n +1 )) lies in B n ( X ). F or λ ∈ B , take λ ′ ∈ B such that fo r any [( g 0 , i 0 ) , . . . , \ ( g j , i j ) , . . . , ( g n +1 , i n +1 )] ∈ C n ( I G ), k c (( g 0 , i 0 ) , . . . , \ ( g j , i j ) , . . . , ( g n +1 , i n +1 )) k λ ≤ k [( g 0 , i 0 ) , . . . , \ ( g j , i j ) , . . . , ( g n +1 , i n +1 )] k λ ′ . Thu s k Φ n ( d n +1 ([( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )])) k λ ≤ ( n + 1) k [( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )] k λ ′ . By the filling norm inequality , for ev ery [( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )] ∈ C n +1 ( I G ) there is an ( n +1)-chain c = c (( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )) with ∂ n +1 c (( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )) = Φ n ( d n +1 ([( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )])), such that for every λ ∈ B there is a n λ ′ ∈ B satisfying k c (( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )) k λ ≤ k [( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 ) k λ ′ . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 25 W e may then pick the fillings c (( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )) to s atisfy the prop erties ( P j,n +1 ), analog ous to those listed ab o ve, and set Φ n +1 (( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )) := c (( g 0 , i 0 ) , . . . , ( g n +1 , i n +1 )). This defines a G -equiv a rian t, linear, B -b ounded chain map Φ n : B C n ( I G ) → B C n ( X ), for 0 ≤ n ≤ k . W e claim Φ n Ψ n = I d, 0 ≤ n ≤ k . First, Φ 0 Ψ 0 ([ v ]) = Φ 0 ([( g v , i ( v ))]) = [ g v x i ( v ) ] = [ v ]. In general, for 1 ≤ n ≤ k , Φ n Ψ n ([ v 0 , . . . , v n ]) = Φ n ([( g 0 , i ( v 0 )) , . . . , ( g i , i ( v n ))]) = c (( g 0 , i ( v 0 )) , . . . , ( g i , i ( v n ))) = [ v 0 , . . . , v n ] as { g 0 x i ( v 0 ) = v 0 , . . . , g n x i ( v n ) = v n } are the vertices of the n -simplex [ v 0 , . . . , v n ]. F or 0 ≤ n < k , consider the map S n : B C n ( X ) → B C n +1 ( X ) given by the comp osition S n = Φ n +1 s n Ψ n . A routine calculation shows ∂ n 1 S n + S n − 1 ∂ n = Φ n Ψ n . Th us { S n } is a c hain co n tra c tion. Mor eo ver, as Φ, Ψ, and s are a ll B - bo unded, so is S , yielding the necessa r y bounded contraction in dimensions n ≤ k , and completing the verification o f ( B 3).  Applying this argument deg r eewise, we o btain the following. Theorem 3 . L et G b e a finitely pr esente d discr ete gr oup of typ e F P ∞ and B a b ounding class. The fol lowing ar e e quivalent. ( B 1 ′ ) ( G, L ) is str ongly B -iso c ohomolo gic al. ( B 2 ′ ) B H ∗ ( G ; V ) → H ∗ ( G ; V ) is surje ct ive for al l b ornolo gic al H B ,L ( G ) -mo dules V . ( B 3 ′ ) G is t yp e B F P ∞ . 4. Rela tive constructions W e show how the results in the previo us section ma y b e “ relativized”. 4.1. Relative H F n and the Brown-Bieri-Ec kmann condi tion. In this sub- section, n will denote an a rbitrary car dinal ≤ ∞ . Given a family of subgroups { H α } α ∈ Λ of G , le t ∆ = ∆( { H α } α ∈ Λ ) := ker M α ∈ Λ Z [ G/H α ] ε → Z ! where ε is the linear extensio n of the map g H α 7→ 1. Then ∆ is a Z [ G ]-mo dule; given a second Z [ G ]-mo dule A , the homolog y of G relative to the family of subgroups { H α } α ∈ Λ with coefficients in A is [ 7 ] H ∗ ( G, { H α } α ∈ Λ ; A ) := T o r Z [ G ] ∗− 1 (∆ , A ) Algebraically , this mak es p erfect se ns e rega rdless o f how the groups intersect. Our ob ject is to establish necess ary and sufficien t conditions for rela tiv e finiteness. A natural s tarting p oint is Lemma 8. Su pp ose that (1) the indexing set Λ is finite, (2) e ach sub gr oup H α is finitely gener ate d, (3) G is finitely pr esente d, (4) T o r Z [ G ] ∗ (∆ , Q Z [ G ]) = 0 for al l 1 ≤ ∗ ≤ ( n − 1) and dir e ct pr o ducts Q Z [ G ] of c opies of Z [ G ] . 26 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Then ∆ is typ e F P n over Z [ G ] . Pr o of. Co nsider the commu ting diagram 0   / / / / ⊕ α ∈ Λ Z [ G ] = / / p 1     ⊕ α ∈ Λ Z [ G ] p 2     ∆ / / / / ⊕ α ∈ Λ Z [ G/H α ] ε / / / / Z where o n each summand p 1 is the natural pro jection Z [ G ] ։ Z [ G/H α ] a nd p 2 = ε ◦ p 1 . Denote ⊕ α ∈ Λ Z [ G/H α ] by E , Z by B , and s et P ( E ) := ⊕ α ∈ Λ Z [ G ], Ω( E ) := ker( p 1 ), Ω( B ) := ker( p 2 ). By the Snak e Le mma , the ab o ve diagr am yields a shor t- exact sequence Ω( E ) ֌ Ω( B ) ։ ∆. Now co nsider the pull-back diagra m Ω( E )     Ω( E )     P ( E , ∆)     / / / / P ( E )     ∆ / / / / E There is a natural iso mo rphism of Z [ G ]-mo dules P ( E , ∆) ∼ = Ω( B ), fro m which w e conclude the exis tence of a short-exact se quence (19) Ω( B ) ∼ = P ( E , ∆) ֌ ∆ ⊕ P ( E ) ։ E Conditions 1. and 2. imply E is finitely-presented ov e r Z [ G ], and condition 3. implies Ω( B ) is finitely-pr esen ted ov er Z [ G ]. By ( 19 ), ∆ ⊕ P ( E ) is finitely-pr esen ted ov er Z [ G ]. But P ( E ) is a finitely- generated free mo dule over Z [ G ] (b y 1. ), so ∆ itself mu st b e finitely-presented ov er Z [ G ]. The result now follo ws from Pr op. 1.2 of [ 6 ].  Definition 7. The gr oup G is typ e F P n +1 (r esp. F F n +1 ) r el. { H α } α ∈ Λ if the Z [ G ] -mo dule ∆ is typ e F P n (r esp. F F n ) over Z [ G ] . As usual, if we ar e not concerned with cons tr ucting finite resolutions but only ones which are finite-dimensional through the given range, the conditions F F n and F P n agree. W e co nsider an alternative definition, whic h will provide the bridge b etw een the algebraic a nd top ologica l setting. Definition 8. A r esolution of Z over Z [ G ] r elative t o a family of sub gr oups { H α } α ∈ Λ is a pr oje ctive r esolution P ∗ of Z over Z [ G ] satisfying P m = ⊕ α ∈ Λ I nd G H α ( W α m ) ⊕ Q m wher e for e ach α ∈ Λ , W α ∗ is a pr oje ctive r esolution of Z over Z [ H α ] and I nd G H α ( W α ∗ ) ֒ → P ∗ is an inclusion of c omplexes. Then G is typ e g F P n , re sp e ctively typ e g F F n , if Q m is finitely gener ate d pr oje ctive, r esp e ct ively fr e e over Z [ G ] , for al l fin it e m ≤ n . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 27 Again, when the reso lutions ar e allow ed to b e infinite, g F P n and g F F n are equiv- alent conditions. Lemma 9. If G is typ e F P n r elative t o { H α } α ∈ Λ , it is typ e g F P n r elative t o { H α } α ∈ Λ . If c onditions 1. - 3. in L emma 8 ar e satisfie d, then t he c onverse is true. Pr o of. Le t R ∗ be a re s olution of ∆ ov er Z [ G ], and let S ∗ be the Z [ G ]-r esolution ⊕ α ∈ Λ I nd G H α ( W α ∗ ) of ⊕ α ∈ Λ Z [ G/H α ] where W α ∗ is a Z [ H α ]-resolution of Z for each α ∈ Λ. Let f ∗ : R ∗ → S ∗ be a map of Z [ G ]-re s olutions co vering the inclusio n ∆ ֌ ⊕ α ∈ Λ Z [ G/H α ], and let M ( f ∗ ) b e the algebra ic mapping cone o f f ∗ . Then M ( f ∗ ) provides a resolution o f Z o ver Z [ G ] rel. { H α } α ∈ Λ where Q m = R m − 1 . Hence t yp e F P n rel. { H α } α ∈ Λ implies t ype g F P n rel. { H α } α ∈ Λ . Now supp ose G is t ype g F P n rel. { H α } α ∈ Λ . As w e hav e seen, the first three conditions of Lemma 8 imply ∆ is finitely-presented. Let T ∗ be a Z [ G ]-resolution of G rel. { H α } α ∈ Λ . Then S ∗ is a sub complex of T ∗ , with the inclusion S ∗ ֒ → T ∗ cov er ing the pro jection ε : ⊕ α ∈ Λ Z [ G/H α ] ։ Z . The quotient co mplex T ∗ /S ∗ = Q ∗ = { Q m } satisfies the prop ert y T or Z [ G ] ∗ (∆ , A ) = H G ∗ +1 ( Q ∗ ⊗ A ) = H ∗ +1 ( Q ∗ ⊗ Z [ G ] A ) If A = Q Z [ G ] and Q m is finitely-generated pro jective for finite m ≤ n , then T or Z [ G ] k  ∆ , Y Z [ G ]  = H G k +1  Q ∗ ⊗ Y Z [ G ]  = H k +1  n Y Q m o m ≥ 0  = 0 for all finite k ≤ ( n − 1). By Lemma 8 this co mpletes the pro of.  W e now consider the topolo gical analogue. Let { A α } α ∈ Λ be a family of sub- complexes of a complex X , and let A = S α ∈ Λ A α ⊆ X . W e say that X is type H F n relative to { A α } α ∈ Λ if X/ A ≃ Y with Y having finitely many cells (or s im- plices) in each finite dimension m ≤ n . If { H α } α ∈ Λ is a family of subgroups of G , then whenever Λ contains mor e than one ele men t ` B H α will not be a subspa ce of the standard mo del for B G , as the clas s ifying spa ces B H α all contain the common basep o in t (and more if the int ersections ar e non- trivial). Howev er, the Z [ G ]-mo dule ∆ is mo deled o n the disjoint union of the c la ssifying space s { B H α } . T o a ccommoda te this, we will need a different mo del for B G . Recall that if T is a disc r ete set, o ne may form the free simplicial set S • ( T ) g enerated by T , with S m ( T ) := T m +1 ; (20) ∂ i ( t 0 , t 1 , . . . , t n ) = ( t 0 , t 1 , . . . , b t i , . . . , t n ); (21) s j ( t 0 , t 1 , . . . , t n ) = ( t 0 , t 1 , . . . , t j , t j , . . . , t n ) (22) In other w ords, the face and degener acy maps are given by deletion and rep etition. An y element of T ca n be used to define a simplical contraction of S • ( T ), yielding S • ( T ) ≃ ∗ for all sets T . Moreov er , if T is a free G -set, then the diag onal action of G makes S • ( T ) a s implicial free G -set, hence a simplicial mo del for a universal contractible G -spac e . The standard homogeneo us bar resolutio n of G - E G - arises when o ne takes T = G with left G -action given by m ultiplication. 28 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Definition 9. F or an indexing set Λ , let G (Λ) := a α ∈ Λ G with G -action given by left mu ltipli c ation on e ach c omp onent. Then E G (Λ) := S • ( G (Λ)) B G (Λ) := E G (Λ) /G Again, if { T α } α ∈ Λ is a collection of G -sets, there is an eviden t inclusion of sim- plicial G -sets (23) a α ∈ Λ S • ( T α ) i Λ ֒ → S • a α ∈ Λ T α ! which is both equiv aria n t and functor ial. Definition 10 . A gr oup G is of typ e H F n r elative to a family of su b gr oups { H α } α ∈ Λ ( n ≤ ∞ ) if B G (Λ) is of t yp e H F n r elative to the image of ` α ∈ Λ B H α under the c om- p osition a α ∈ Λ B H α ֒ → a α ∈ Λ B G i Λ ֒ → B G (Λ) Prop osition 2. Assum e the indexing set Λ is finite. If G is typ e H F n r elative to { H α } α ∈ Λ , t hen it is typ e g F F n r elative to { H α } α ∈ Λ . Pr o of. F or each α , fix a simplicia l complex X H α ≃ B H α . The conditio n o n G implies we may c o nstruct a simplicia l mo del X G ≃ B G by a dding simplices (i.e., cells) to ` α ∈ Λ X H α in such a wa y that through each finite dimension m with 0 ≤ m ≤ n the n umber of simplices a ttac hed is finite. Let e X G ≃ E G b e the universal cov er of X G , and p : e X G → X G the cov ering map. T aking P ∗ = C ∗ ( e X G ) g ives the desired resolution, with I nd G H α ( W ∗ ) = C ∗ ( p − 1 ( X H α )) and Q n the free C [ G ]-submo dule of P n spanned o ver C by those n -cells not in p − 1  ` α ∈ Λ X H α  .  Putting it all together , we may summariz e the situation a s Theorem 4. If the c onditions 1. - 4. of L emma 8 ar e satisfie d, then the fol lowi ng ar e e qu ivalent • G is typ e H F n r elative t o { H α } α ∈ Λ , • G is typ e g F F n r elative t o { H α } α ∈ Λ , • G is typ e F F n r elative t o { H α } α ∈ Λ . Pr o of. The se c o nd and third prop erties are equiv alent by Lemma 9 , and the firs t prop ert y implies the s e cond by the previo us pro position. The conv er se to P ropo - sition 2 , in the presence of conditions 1. through 4. , follo ws by the sa me metho d of geo metrically rea lizing the resolution a s in the classica l pro of of the Eilenberg- Ganea-W all theorem (compare Thm. 7.1, Chap. VII I of [ 10 ]).  Recall from [ 11 ] that a direct system o f g roups { A β } is sa id to be essentially trivial if for each β 1 there is a β 2 ≥ β 1 such that the ma p A β 1 → A β 2 is trivial. B -BOUNDED COHOMOLOGY AND APPLICA TIONS 29 Definition 11. A filtr ation of E G (Λ) of finite n - typ e r elative to { H α } α ∈ Λ is an incr e asing filt r ation of E G (Λ) by a dir e ct syst em of sub c omplexes { X β } satisfying • lim − → β X β = E G (Λ) , • X ∅ := p − 1  ` α ∈ Λ B H α  ⊆ X β for al l β , • F or e ach β , X β /X ∅ c ontains finitely many G -orbits in fi nite dimensions ≤ n . Theorem 5. Supp ose ther e exists a filtr ation { X β } of finite n -typ e of E G r elative to { H α } α ∈ Λ . Assume c onditions 1. - 4. of L emma 8 . If t he dir e cte d system { H ∗ ( X β ) } is essential ly trivial for al l finite ∗ ≤ n , then ∆ is typ e F P n − 1 over Z [ G ] . Pr o of. Le t R = Z [ G ]. Then fo r all finite m with 1 ≤ m ≤ ( n − 1) T or R m  ∆ , Y R  ∼ = T or R m  R, ∆ ⊗ Y R  = H G m  E G (Λ); ∆ ⊗ Y R  ∼ = H G m +1  E G, X ∅ ; Y R  ∼ = lim − → β H G m +1  X β /X ∅ ; Y R  ∼ = lim − → β Y H G m +1 ( X β /X ∅ ; R ) = lim − → β Y H m +1 ( X β /X ∅ ) = 0 with the la s t equality following by Lemma 2.1 of [ 11 ].  Note that unlike Br o wn’s condition in the a bsolute ca se (where we ar e not working relative to a family of subgr o ups), the quotient space E G (Λ) / X ∅ has the homotopy-t y pe of a wedge W S 1 , with H 1 ( W S 1 ) = ∆. This non-contractibilit y in dimension 1 makes verifying the ess en tial tr ivialit y of relative filtrations problem- atic. 4.2. Relative Dehn functions . There is a na tur al a lgebraic wa y to define the relative Dehn functions of G with resp ect to s ome finite family of subgroups . As befo re, we write ∆ for the kernel of the augmentation map ǫ : L α ∈ Λ Z [ G/H α ] → Z , where H = { H α | α ∈ Λ } . A t ypical element of L α ∈ Λ Z [ G/H α ] has the form P g ∈⊔ G/H α λ g g [ H α g ]. Th us to define the w eighted structure on ∆ ⊂ L α ∈ Λ Z [ G/H α ], it is enoug h to define the weigh t of a generator , g [ H α ]. Set w ( g [ H α ]) = min h ∈ H α L ( g h ), where L is the length function equipp ed on G . If G is of t yp e F F ∞ relative to H , there is a free resolution of ∆ ov er Z [ G ]: . . . → R 2 → R 1 → R 0 → R − 1 := ∆ → 0 which is finitely ge nerated Z [ G ]-mo dule in each dimensio n o n ge ne r ating set S n = { s 1 ,n , . . . , s k n ,n } . Fixing the weigh t of each generator to b e 1, we extend this to a weigh t function o n R n in the usua l wa y b y w  X λ i g i s j i ,n  := X | λ i | ( L ( g i ) + w ( s j i ,n )) = X | λ i | ( L ( g i ) + 1) 30 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Up to linear equiv alence, this definition is independent of the initial weigh ting given to the gener ating set. N ow choos e a Z -linear s plitting of this res olution { s n : R n → R n +1 } n ≥− 1 ; as s ociated to this contraction are its Dehn functions, which we refer to as the relative Dehn functions of G with respect to H . Again, up to linea r equiv alence, the definition of the Dehn functions is independent of the choice of linear s plitting. Definition 12 . The r elative Dehn functions of G with r esp e ct to H ar e B -b oun de d if ther e is typ e F F ∞ r esolution of ∆ over Z [ G ] such that e ach splitting s n is b ou n de d by an element of B . Lemma 10. S upp ose L  B . If t he r elative Dehn functions of G with re sp e ct to H ar e B - b ounde d for a p articular typ e F F ∞ r esolution, then they ar e B -b oun de d for al l typ e F F ∞ r esolutions. This is a rela tive version of the statemen t that the Dehn functions of a group G do es not dep end on which type H F ∞ classifying spac e is used in their construction, up to equiv alenc e . Since G is of relative type F F ∞ , it is also of relative type g F F ∞ , with resp ect to H . As abov e, this gives a pro jectiv e resolution of Z o ver Z [ G ] of the form P m = M α ∈ Λ I nd G H α ( W α m ) ⊕ Q m where for each α ∈ Λ, W α ∗ is a pro jective resolution of Z over Z [ H α ], and Q m is a finitely generated free Z [ G ]-mo dule. At eac h level, I nd G H α ( W α m ) is a direct s ummand of P m . T aking the quotient b y them in each deg ree yields the sequence . . . → Q 3 → Q 2 → Q 1 ։ Q 0 = 0 which is exa ct ab o ve dimension 1, and for which the cokernel of Q 2 → Q 1 is ∆. Thu s . . . → Q 3 → Q 2 → Q 1 → ∆ → 0 is a type F F ∞ resolution of ∆ ov er Z [ G ]. Because the homology of bo th P ∗ and Q ∗ v anish ab ove dimension 1, o ne can in that ra nge c onstruct a chain contraction { s P n : P n → P n +1 } for which the co mpo- sition s Q n := Q n ֒ → P n s P n − → P n +1 ։ Q n +1 yields a chain contraction of Q ∗ for ∗ > 1. This splitting, spliced together with a weigh t-minimizing section s Q 0 : ∆ → Q 1 of the pro jection Q 1 ։ ∆, one ca n define the top ological relative Dehn functions of G relative to H to b e the Dehn functions asso ciated to the linear contraction { s Q n } n ≥ 0 . W e say the top ological Dehn functions are B - bounded if the Dehn functions of { s Q n } n ≥ 0 are B -bo unded. Lemma 11. Supp ose G is of typ e F F ∞ r elative to H . G has B - b ounde d algebr aic r elative Dehn functions with r esp e ct to H if and only if it has B -b oun de d top olo gic al r elative Dehn fu n ctions with r esp e ct to H This follows immediately from Lemma 10 . The term “topo logical rela tiv e Dehn function” is justified by the following in- terpretation of them. Assume G is H F ∞ relative to H , as b e fore. Star t with a relative homo logy cycle in x ∈ Z n ( E G (Λ) , E ( H )). Then ∂ ( x ) ∈ C n − 1 ( E ( H )). Each part of ∂ ( x ) lying in a connected c omponent of E ( H ) can be coned off, yielding an B -BOUNDED COHOMOLOGY AND APPLICA TIONS 31 absolute cyc le x ′ ∈ Z n ( E G (∆)). Cho ose a weigh t-minimizing y ∈ C n +1 ( E G (∆)) with ∂ ( y ) = x ′ , and take its weigh t relative to the subspace E ( H ); i.e., only total the weights of those ( n + 1)-simplices used to construct z which do no t lie in E ( H ). The resulting Dehn function computed using this constructio n agree s (up to the usual equiv alence o f Dehn functions) with the one derived from { s Q n } n ≥ 0 . One can vie w the nonexistence of a relative Dehn function in a particula r degree as an obstr uction to completing the type F F ∞ resolution of ∆, . . . → Q 3 → Q 2 → Q 1 → ∆ → 0 to yield a type F F ∞ bo rnological res olution of ∆ B (defined below). . . . → B Q 3 → B Q 2 → B Q 1 → ∆ B → 0 Although there is alwa ys a b ounded sectio n ∆ → Q 1 , the obs truction to cons tr uct- ing a bounded se c tion Q 1 → Q 2 is, in general, nontrivial in the unw eig h ted setting. The r elativ e Dehn functions for n > 1 do not suffer the same is sue. 4.3. Relative B -b ounded cohomol ogy . W e construct a rela tiv e B -bo unded co - homology theo r y , which closely mirror s the construc tio n of relative gro up co homol- ogy in [ 4 , 7 ]. Let G b e a discrete group a nd let H = { H α | α ∈ Λ } be a finite collection of s ubgroups o f G . L et G/ H b e the disjoint union F α ∈ Λ G/H α . F or a subgr o up H of G let C [ G/H ] b e the C -vector space with bas is the left c o sets G/H . Let C [ G/ H ] = ⊕ α ∈ Λ C [ G/H α ] which will b e identified as finitely supp orted functions G/ H → C . Denote the kernel of the augmentation ε : C [ G/ H ] → C b y ∆. Definition 13. The r elative c ohomolo gy of a discr ete gr oup G , with r esp e ct to a c ol le ction H of sub gr oups of G with c o efficients in a C [ G ] -mo dule A, is given by H k ( G, H ; A ) = Ext k − 1 C [ G ] (∆ , A ) . Denote by H k ( H ; A ) = Q α ∈ Λ H k ( H α ; A ). The definition o f rela tiv e cohomolo gy yields the following conse q uence, proved in [ 4 ] for the ca se of a single subgr oup and in [ 7 ] for man y subgroups. Theorem 6 (Auslander, Bieri-Eckmann) . L et G and H b e as ab ove. F or any C [ G ] -mo dule A ther e is a long exact se quenc e: . . . → H k ( G ; A ) → H k ( H ; A ) → H k +1 ( G, H ; A ) → H k +1 ( G ; A ) → . . . . The leng th- function L on G induces a weigh t, w , on the cosets G/H given by w ( g H ) = min h ∈ H L ( g h ). With this weigh t, define the following bor nological H B ,L ( G )-mo dule: H B ,w ( G/H ) = { f : G/H → C | ∀ φ ∈B X x ∈ G/H | f ( x ) | φ ( w ( x )) < ∞} . This is a F rechet spa ce in the norms giv en b y k f k φ = X x ∈ G/H | f ( x ) | φ ( w ( x )) . Let H = { H α | α ∈ Λ } b e a finite collection of subg roups of G , and define H B ,w ( G/ H ) = { f : G/ H → C | ∀ φ ∈B X x ∈ G/ H | f ( x ) | φ ( w ( x )) < ∞} . 32 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY As H is a finite family of subgro ups, H B ,w ( G/ H ) = ⊕ α ∈ Λ H B ,w ( G/H α ). The augmentation map ε : H B ,w ( G/ H ) → C is given b y ε ( f ) = P x ∈ G/ H f ( x ). As ε ( f ) ≤ k f k 1 , ε is a b ounded ma p. Denote the augmentation kernel by ∆ B = ker ε . Definition 14. The r elative B -b oun de d c oho molo gy of a discr ete gr oup G , with r esp e ct to a c ol le ction H of sub gr oups of G with c o efficients in an H B ,L ( G ) -mo dule A, is given by B H k ( G, H ; A ) = Ex t k − 1 H B ,L ( G ) (∆ B , A ) , wher e this Ext ∗ H B ,L ( G ) ( · , A ) functor is taken over the c ate gory of b ornolo gic al H B ,L ( G ) -mo dules. As in the absolute cohomology theory , there is a comparison homomorphism B H ∗ ( G, H ; V ) → H ∗ ( G, H ; V ) for any b ornolo g ical H B ,L ( G )-mo dule V . Definition 15. L et G b e a discr ete gr oup with length function L , and let H b e a fi nite c ol le ction of sub gr oups, and let V b e a H B ,L ( G ) -mo dule. We say G is r elatively B -iso c ohomolo gic al to H with r esp e ct to V (abbr. V - B RIC) if the r elative c omp arison B H ∗ ( G, H ; V ) → H ∗ ( G, H ; V ) is an isomorphism of c ohomolo gy gr oups in al l de gr e es. Similarly G is re latively B -iso c ohomolo gic al t o H (abbr. B -R IC) if it is C - B RIC, and G is str ongly r elatively B - iso c ohomo lo gic al to H (abbr. B -SRIC) if it is V - B RIC fo r al l H B ,L ( G ) -mo dules V . If G is a group a nd H is a subgroup, there is an isomorphism: Ext ∗ C [ G ] ( C [ G/H ] , C ) ∼ = Ext ∗ C [ H ] ( C , C ) . A first step in extending relative co homology to the B -b ounded fr amew o rk will be the follo wing ana logue. Lemma 12. L et G b e a gr oup with length function L , H a sub gr oup of G e quipp e d with t he r est r icte d length function, and B a m ultiplic ative b ounding class. F or any b ornolo gic al H B ,L ( G ) -mo dule A , ther e is an isomorphism: Ext ∗ H B ,L ( G ) ( H B ,w ( G/H ) , A ) ∼ = Ext ∗ H B ,L ( H ) ( C , A ) . Before pr o ving Lemma 12 we first turn o ur atten tion to a few additional r esults which will be necessar y . Lemma 1 3 . L et B b e a mu ltipli c ative b ounding class, and endow the c oset sp ac e G/H with the weight w ( g H ) = min h ∈ H L ( g h ) , and L is the length function on G . Then H B ,L ( G ) ∼ = H B ,w ( G/H ) ˆ ⊗H B ,L ( H ) b oth as b ornolo gic al ve ctor sp ac es and as right H B ,L ( H ) -mo dules. Pr o of. Le t R be a sy s tem of minimal length re presen tatives for left cosets of H in G . F or an r ∈ R , the length o f r in G is equal to the length of rH ∈ G/H , so H B ,w ( G/H ) ∼ = H B ,L ( R ) as b ornological vector s paces. F or g ∈ G there is a unique h g ∈ H and r g ∈ R such that g = r g h g . Let Φ : H B ,L ( G ) → H B ,L ( R ) ˆ ⊗H B ,L ( H ) b e defined on ba sis elements by Φ( g ) = ( r g ) ⊗ ( h g ) and extended by linea rit y . F or λ, µ ∈ B , let ν ∈ B such that ν ( n ) ≥ λ (2 n ) µ (2 n ). | Φ( g ) | λ,µ = | ( r g ) ⊗ ( h g ) | λ,µ = λ ( L ( r g )) µ ( L ( h g )) ≤ λ ( L ( g )) µ ( L ( r − 1 g ) + L ( r g h g )) ≤ λ ( L ( g )) µ (2 L ( g )) B -BOUNDED COHOMOLOGY AND APPLICA TIONS 33 ≤ λ (2 L ( g )) µ (2 L ( g )) ≤ ν ( L ( g )) . It follo ws that Φ is b ounded. Conv ers ely let Ψ ′ : H B ,L ( R ) × H B ,L ( H ) → H B ,L ( G ) be defined b y Ψ ′ (( r , h )) = ( rh ). F o r λ ∈ B , let λ ′ ( n ) = λ (2 n ) + 1. By the pro perties of b ounding clas s es, λ ′ ∈ B as well. F or r ∈ R and h ∈ H , se t M r,h = max { L ( r ) , L ( h ) } and m r,h = min { L ( r ) , L ( h ) } . W e hav e : | Ψ ′ ( r , h ) | λ = | ( rh ) | λ = λ ( L ( rh )) ≤ λ ( L ( r ) + L ( h )) ≤ λ (2 M r,h ) ≤ λ ′ ( M r,h ) ≤ λ ′ ( M r,h ) λ ′ ( m r,h ) = λ ′ ( L ( r )) λ ′ ( L ( h )) . As Ψ ′ is b ounded, it ex tends to a bounded Ψ : H B ,L ( R ) ˆ ⊗H B ,L ( H ) → H B ,L ( G ). These are the required bor nological iso morphisms.  Lemma 14. F or any b ounding class B , H B ,L ( G ) ˆ ⊗ H B ,L ( H ) C ∼ = H B ,w ( G/H ) , wher e H is endowe d with t he r estricte d length function and G/H is endowe d with the weight w . Pr o of. B y [ 26 ], if E = H ˆ ⊗ A then E ˆ ⊗ A F ∼ = H ˆ ⊗ F . Appealing to the pre v ious lemma we obtain the following. H B ,L ( G ) ˆ ⊗ H B ,L ( H ) C ∼ =  H B ,w ( G/H ) ˆ ⊗H B ,L ( H )  ˆ ⊗ H B ,L ( H ) C ∼ = H B ,w ( G/H ) ˆ ⊗ C ∼ = H B ,w ( G/H ) .  Pr o of of L emma 12 . Consider . . . → H B ,L ( H ) ˆ ⊗ 3 → H B ,L ( H ) ˆ ⊗H B ,L ( H ) → H B ,L ( H ) → C → 0 where the b oundary map d n +1 : H B ,L ( H ) ˆ ⊗ n +1 → H B ,L ( H ) ˆ ⊗ n is given by d n +1 ( h 0 , h 1 , . . . , h n ) = ( h 0 h 1 , h 2 , . . . , h n ) − ( h 0 , h 1 h 2 , h 3 , . . . , h n ) + . . . + ( − 1) n − 1 ( h 0 , h 1 , . . . , h n − 1 h n ) +( − 1) n ( h 0 , . . . , h n − 1 ) and d 1 : H B ,L ( H ) → C , g iv en b y d 1 ( h 0 ) = 1, is the a ugmen ta tio n. There is a bo unded contracting homotopy given b y s n +1 ( h 0 , . . . , h n ) = (1 G , h 0 , . . . , h n ), where 1 G is the identit y element o f G , and s 0 : C → H B ,L ( G ) is g iven by s 0 ( z ) = z (1 G ). Ext ∗ H B ,L ( H ) ( C , A ) is the cohomology of the co c ha in complex Hom bdd H B ,L ( H ) ( H B ,L ( H ) , A ) → Hom bdd H B ,L ( H ) ( H B ,L ( H ) ˆ ⊗H B ,L ( H ) , A ) → Hom bdd H B ,L ( H ) ( H B ,L ( H ) ˆ ⊗ 3 , A ) → . . . . 34 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY By [ 26 ], for B a bor nological a lgebra, E a n y bo rnological space, and F any bor nolog- ical left B -mo dule, Hom bdd B ( B ˆ ⊗ E , F ) ∼ = Hom bdd ( E , F ). Thus Ext ∗ H B ,L ( H ) ( C , A ) is the cohomology of Hom bdd ( C , A ) → Hom bdd ( H B ,L ( H ) , A ) → Hom bdd ( H B ,L ( H ) ˆ ⊗ 2 , A ) → . . . . T ensor ing each of the left H B ,L ( H ) modules H B ,L ( H ) ˆ ⊗ n by H B ,L ( G ) o ver H B ,L ( H ) yields . . . → H B ,L ( G ) ˆ ⊗ H B ,L ( H ) ( H B ,L ( H ) ˆ ⊗H B ,L ( H )) → H B ,L ( G ) ˆ ⊗ H B ,L ( H ) H B ,L ( H ) → H B ,L ( G ) ˆ ⊗ H B ,L ( H ) C → 0 . As H B ,L ( G ) ˆ ⊗ H B ,L ( H ) C ∼ = H B ,w ( G/H ), this reduces to . . . → H B ,L ( G ) ˆ ⊗ H B ,L ( H ) ( H B ,L ( H ) ˆ ⊗H B ,L ( H )) → H B ,L ( G ) ˆ ⊗ H B ,L ( H ) H B ,L ( H ) → H B ,w ( G/H ) → 0 . The bor nological isomor phism H B ,L ( G ) ˆ ⊗ H B ,L ( H ) H B ,L ( H ) ˆ ⊗ n +1 ∼ = H B ,L ( G ) ˆ ⊗H B ,L ( H ) ˆ ⊗ n , given b y [ 26 ], sho ws that this is . . . → H B ,L ( G ) ˆ ⊗H B ,L ( H ) ˆ ⊗ 2 → H B ,L ( G ) ˆ ⊗H B ,L ( H ) → H B ,L ( G ) ։ H B ,w ( G/H ) → 0 . The map δ n +1 : H B ,L ( G ) ˆ ⊗H B ,L ( H ) ˆ ⊗ n → H B ,L ( G ) ˆ ⊗H B ,L ( H ) ˆ ⊗ n − 1 is given by δ n +1 ( g , h 0 , . . . , h n ) = ( g h 0 , h 1 , . . . , h n ) − ( g , h 0 h 1 , h 2 , . . . , h n ) + . . . + ( − 1) n ( g , h 0 , . . . , h n − 1 h n ) +( − 1) n +1 ( g , h 0 , . . . , h n − 1 ) while the map δ 1 : H B ,L ( G ) → H B ,w ( G/H ) is given by δ 1 ( g ) = ( g H ). A b ounded contracting homotopy s ′ n is constructed as follows. The ma p s ′ 0 : H B ,w ( G/H ) → H B ,L ( G ) is given b y s ′ 0 ( g H ) = ( r g ), w her e r g is the fixed minimal length r epresent a- tive o f the cos e t g H in R as above. The map s ′ 1 : H B ,L ( G ) → H B ,L ( G ) ˆ ⊗H B ,L ( H ) is given by s ′ 1 ( g ) = ( r g , h g ), where r g ∈ R , h g ∈ H , and g = r g h g . The same F or n > 1, s ′ n ( g , h 0 , . . . , h n − 2 ) = ( r g , h g , h 0 , . . . , h n − 2 ). As this is a pro jective reso lution of H B ,w ( G/H ) o ver H B ,L ( G ), Ext ∗ H B ,L ( G ) ( H B ,w ( G/H ) , A ) can be calculated as the cohomo logy of Hom bdd H B ,L ( G ) ( H B ,L ( G ) , A ) → Hom bdd H B ,L ( G ) ( H B ,L ( G ) ˆ ⊗H B ,L ( H ) , A ) → Hom bdd H B ,L ( G ) ( H B ,L ( G ) ˆ ⊗H B ,L ( H ) ˆ ⊗ 2 , A ) → . . . . As Hom bdd H B ,L ( G ) ( H B ,L ( G ) ˆ ⊗H B ,L ( H ) ˆ ⊗ n , A ) ∼ = Hom bdd ( H B ,L ( H ) ˆ ⊗ n , A ), this is also the same a s the cohomology o f Hom bdd ( C , A ) → Hom bdd ( H B ,L ( H ) , A ) → Hom bdd ( H B ,L ( H ) ˆ ⊗ 2 , A ) → . . . .  Denote b y B H k ( H ; A ) = Q α ∈ Λ B H k ( H α ; A ). Theorem 7. L et G and H b e as ab ove, and let A b e a b ornolo gic al H B ,L ( G ) -mo dule. F or any multiplic ative b ounding class B , ther e is a long exact se quenc e: . . . → B H k ( G ; A ) → B H k ( H ; A ) → B H k +1 ( G, H ; A ) → B H k +1 ( G ; A ) → . . . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 35 wher e for e ach H α ∈ H , H α is given the length function r estricte d fr om G and G/ H is given the minimal weighting function w as ab ove. Pr o of. The following shor t exact sequence admits a b ounded C -splitting. 0 → ∆ B → H B ,w ( G/ H ) ε → C → 0 Applying the b ornolo gical Ext ∗ H B ,L ( G ) ( · , A ) functor , yields the long exact se- quence . . . → Ex t k H B ,L ( G ) ( C , A ) → Ext k H B ,L ( G ) ( H B ,w ( G/ H ) , A ) → Ext k H B ,L ( G ) (∆ B , A ) → Ext k +1 H B ,L ( G ) ( C , A ) → . . . . Making use of the isomorphisms, Ext k H B ,L ( G ) ( H B ,w ( G/ H ) , A ) = Ext k H B ,L ( G ) ( ⊕ α ∈ Λ H B ,w ( G/H α ) , A ) ∼ = Y α ∈ Λ Ext k H B ,L ( G ) ( H B ,w ( G/H α ) , A ) ∼ = Y α ∈ Λ Ext k H B ,L ( H α ) ( C , A ) one obtains the exact sequence . . . → Ex t k H B ,L ( G ) ( C , A ) → Y α ∈ Λ Ext k H B ,L ( H α ) ( C , A ) → Ext k H B ,L ( G ) (∆ B , A ) → Ext k +1 H B ,L ( G ) ( C , A ) → . . . . By definition E xt k H B ,L ( G ) ( C , A ) = B H k +1 ( G ; A ), and Ext k H B ,L ( G ) (∆ B , A ) = B H k +1 ( G, H ; A ).  Corollary 3. L et G b e a finitely gener ate d gr oup with length fun ction L , B a multiplic ative b ounding class, and H = { H α | α ∈ Λ } a finite family of sub gr oups. Supp ose that ther e is a H B ,L ( G ) -mo dule V such that e ach H α is V - B IC, in the length function r estricte d fr om G . If G is V - B RIC to H , then G is V - B IC. In p articular, if e ach H α is B -SIC and G is B -S RIC to H , t hen G is B -S IC. Pr o of. The comparison map yields a commutativ e diagra m with top row the long exact sequence from Theorem 7 and the bo ttom r ow the long exact sequence from Theorem 6 . The result follows fro m the five-lemma.  The notion of ‘niceness’ defined above has an ob vio us extensio n to free resolutio ns of mo dules other than Z ov er Z [ G ]. The following generaliz a tion of Lemma 1 to the relative setting is stra ig h tforward. Lemma 15. L et G b e a gr oup e quipp e d with wor d-length function L , R ∗ a k -nic e r esolution of M over Z [ G ] , T ∗ = R ∗ ⊗ C , and B and B ′ b ounding classes. Denote by B T ∗ the c orr esp onding F r e chet c ompletion of T ∗ with r esp e ct to the b ounding class B , as define d ab ove. F urther s upp ose t hat the weighte d Dehn funct ions { d w, n R } ar e B ′ -b ounde d in dimensions n < k , that B is a right B ′ -class, and that B  L . Then ther e exists a b ou n de d chain n ul l-homotopy { s n +1 : B T n → B T n +1 } k>n ≥ 0 , imply ing B T ∗ is a c ontinuous r esolut ion of B M over H B ,L ( G ) thr ough dimension k . Her e, B M denotes the c ompletion of M ⊗ C via the b ounding class B . This yields a suitable complex from which we ma y ca lculate b ounded rela tiv e cohomolog y o f G with resp ect to H . 36 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Theorem 8. Su pp ose the finitely gener ate d gr oup G is H F ∞ r elative t o the finite family of fi nitely gener ate d su b gr oups H . The fol lowing ar e e quivalent: (1) The r elative Dehn functions of G r elative to H ar e e ach B - b ounde d. (2) G is B -SRIC with r esp e ct to H . (3) The c omp arison map B H ∗ ( G, H ; A ) → H ∗ ( G, H ; A ) is surje ctive for al l b ornolo gic al H B ,L ( G ) -mo dules A . Pr o of. F or (1) implies (2), suppose A is a b ornologic a l H B ,L ( G )-mo dule, and let R ∗ := . . . → R 2 → R 1 → R 0 → R − 1 = ∆ Z → 0 be a nice type F F ∞ resolution of ∆ Z , the integral augmentation kernel, over Z [ G ], and let T ∗ := . . . → T 2 → T 1 → T 0 → T − 1 = ∆ → 0 be given by T n = R n ⊗ C . Here ∆ is the c o mplex aug men tation kernel. T ∗ is a t ype F F ∞ resolution of ∆ ov er C [ G ]. As the relative Dehn functions are B - bo unded, the pr e vious lemma gives that B T ∗ is a b ornologica lly pro jective res olution of ∆ B ov er H B ,L ( G ). Let V n be the complex vector space with one basis elemen t for each generator o f T n ov er C [ G ]. Ther e are isomor phisms T n ∼ = C [ G ] ⊗ V n and B T n ∼ = H B ,L ( G ) ˆ ⊗ V n . Apply Hom C [ G ] ( · , A ) to the deleted r esolution T ∗ yields a cochain complex with terms of the form Ho m C [ G ] ( T n , A ). Applying Hom bdd H B ,L ( G ) ( · , A ) to the deleted res o- lution B T ∗ yields a co chain complex with terms o f the form Hom bdd H B ,L ( G ) ( B T n , A ). Hom C [ G ] ( T n , A ) ∼ = Hom C [ G ] ( C [ G ] ⊗ V n , A ) ∼ = Hom( V n , A ) ∼ = Hom bdd H B ,L ( G ) ( H B ,L ( G ) ˆ ⊗ V n , A ) ∼ = Hom bdd H B ,L ( G ) ( B T n , A ) As A was ar bitrary we obta in that G is B -SRIC with respect to H . The implication (2) implies (3) is obvious. F or (3) implies (1), follow the pro of the implication ( B 2) implies ( B 1) of Theor em 3 with the fo llo wing mo difications. Replace the absolute co cycles and b oundaries, by the relative co cycles and bound- aries. The ar gumen t applies nearly verbatim.  F or the remainder of the sec tion, we supp ose tha t G is a finitely pr e sen ted group which acts co compactly without in version on a contractible complex X , with finite edge s ta bilizers and finitely genera ted vertex stabilizers G σ . The higher weigh ted Dehn functions o f X b ound the topolog ical relative Dehn functions of G with r espect to the { G σ } . Applying Theor em 8 w e o btain the following. Theorem 9. Supp ose al l of t he higher weighte d Dehn functions of C ∗ ( X ) ar e B - b ounde d. Then G is B -S R IC with r esp e ct to the { G σ } . T o use this r esult e ffectively , we m ust b e able to determine how the res tricted length function on G σ behaves, whe n compared to the usual w ord-length function on G σ . Lemma 16. Su pp ose t he first unweighte d ge ometric Dehn fu n ction of X is B - b ounde d. Then the standar d wor d-length function, L G σ , on G σ , for every σ , is B -e quivalent to t he length function on G σ , induc e d by t he r estriction of L G to G σ . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 37 Sp e cific al ly, ther e exists a ν ∈ B such that L G σ ( g ) ≤ ν ( L G ( g )) for al l g ∈ G σ and al l σ . Pr o of. The finite edge s tabilizers imply that the first re la tiv e Dehn function ( in the meaning of Osin [ 37 ] ), is equiv a len t to the first Dehn function of X , b y [ 9 ]. Thus it is B -b ounded. By Lemma 5.4 of [ 37 ], the dis tortion of each H ∈ H is b ounded by the relative Dehn function. Thus ea ch H is at most B -distorted.  The following is a gene r alization of a r e sult in [ 23 ], whic h states that if a finitely generated gro up G is relatively h yp erb olic to a family of finitely genera ted sub- groups H , a nd if each H ∈ H is H F ∞ and P -SIC, then G is P -SIC. Corollary 4. Supp ose the finitely gener ate d gr oup G is r elatively hyp erb olic with r esp e ct to the family of finitely gener ate d sub gr oups H . If L  B , then G is B -SRI C with r esp e ct to H . Pr o of. B y Mineyev-Y aman [ 31 ], there is a con tractible h yp erb olic co mplex X on which G acts coco mpactly with finite edg e s tabilizers, a nd vertex stabilizers pre- cisely the H and their translates. Lemma 16 giv es the result.  Corollary 5. Supp ose the finitely gener ate d gr oup G is r elatively hyp erb olic with r esp e ct to the family of finitely gener ate d sub gr oups H , and B is a multiplic ative b ounding class with L  B . F or any b ornolo gic al H B ,L ( G ) -mo dule M , if e ach H ∈ H is M - B IC, then G is M - B IC. In p articular, if e ach H is B -SIC then G is B -SIC. 5. Two spectral sequences in B -bounded cohomol ogy 5.1. The H ochsc hild-Serre sp ectra l sequence. W e b egin with a finiteness re- sult which w a s first prov en for p olynomially bounded coho mo logy in [ 25 ]. Theorem 10. L et ( G, L ) b e V - B IC with r esp e ct to the t rivia l H B ,L ( G ) -mo dule V 6 = 0 . Assume that V is metrizable, with distanc e function d V . Then for e ach n ≥ 0 , B H n ( G ; V ) = H n ( G ; V ) ∼ = k n M V Pr o of. B y contradiction. Fir st, note that the weight function on C ∗ ( E G ) induces a w eight function on H ∗ ( B G ) by w ([ x ]) = min { w ( x ′ ) | [ x ′ ] = [ x ] } . The state- men t that G is V - B IC is then equiv alent to req uiring that for each n ≥ 1 and fo r all [ c ] ∈ H n ( B G ), there exists a φ n ∈ B such that for a ll [ x ] ∈ H n ( B G ) , d V ([ c ]([ x ]) , 0) ≤ φ n ( w ([ x ])). Next, H n ( G ; V ) ∼ = Hom( H n ( G ) , V ) by the Universal Co efficient The- orem. Supp ose H n ( G ; V ) is not a finite sum of co pies of V . This can only hap- pen if H n ( G ) is infinite-dimensional o ver C . Cho ose a coun tably infinite linearly independent set [ x 1 ] , [ x 2 ] , . . . , [ x n ] , . . . of elements in H n ( G ); we nor ma lize ea c h element so that w ([ x i ]) = 1 for e a c h 1 ≤ i . Then for each i , fix a cohomo lo gy class [ c i ] ∈ H n ( G ; V ) with d V ([ c i ]([ x j ]) , 0 ) = δ ij . The set { [ c 1 ] , [ c 2 ] , . . . , [ c m ] , . . . } is a countably infinite gener ating set for a subs pa ce W = Q ∞ 1 C ⊂ H n ( G ; V ). An element of W may b e wr itten as C = ( n 1 , n 2 , n 3 , . . . , n m , . . . ) indicating that the [ c i ]-comp onen t of C is n i [ c i ]. Define a cohomology cla ss [ C f ] by [ C f ] = ( f (1) , f (2) , . . . , f ( n ) , . . . ) ∈ W 38 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY By construction, d V ([ C f ]([ x m ]) , 0 ) = f ( m )[ c m ]([ x m ]) = f ( m ) for each m ≥ 1. Choo sing the function f to be unbo unded (whic h w e can certainly do) makes [ C f ] unbounded on the set of n - dimensional ho mo logy c la sses with weigh t 1. This contradicts the a ssumption tha t ( G, L ) is B -iso cohomolog ic a l with resp ect to V , regar dless of the choice of B .  The following theorem was prov en in [ 34 ] in the con text of p.s. G -modules . It was shown in the b ornolo gical case in [ 3 8 ]. Theorem 11 ([ 38 , 3 4 ]) . L et 0 → ( G 1 , L 1 ) → ( G 2 , L 2 ) → ( G 3 , L 3 ) → 0 b e an extension of gr oups with wor d-length, with G 3 typ e P F P ∞ . Ther e is a b ornolo gic al sp e ctr al se quenc e with E p,q 2 ∼ = P H p ( G 3 ; P H q ( G 1 )) and which c onver ges to P H ∗ ( G ) , wher e ˆ ⊗ is the b ornolo gic al c omplete d pr oje ctive tens or pr o duct. (An “extension of groups with word-length” means L 1 and L 3 are induced by the length function L 2 .) A similar r esult holds with more ge neral b ounding classes and co efficients. Theorem 12 (Serre Spectra l Seq uenc e in B -b ounded cohomology) . L et ( G 1 , L 1 ) ֌ ( G 2 , L 2 ) ։ ( G 3 , L 3 ) b e an extension of gr oups with wor d-length. Supp ose that V is a metrizable (b ornolo gic al) H B ,L 2 ( G 2 ) -mo dule and ( G 3 , L 3 ) is B -SIC. Then ther e exists a sp e ctr al se quenc e E p,q 2 = B H p ( G 3 ; B H q ( G 1 ; V )) ⇒ B H p + q ( G 2 ; V ) Henc e if ( G 1 , L 1 ) is B - SIC, so is ( G 2 , L 2 ) . Pr o of. Le t ( P ∗ , d P ) b e the B -c ompletion of the homogeneous bar resolution of G 2 and T ∗ be the tensor product of P ∗ by C over H B ,L 1 ( G 1 ), T q ∼ = H B ,L 3 ( G 3 ) ˆ ⊗H B ,L 2 ( G 2 ) ˆ ⊗ q . By h yp othesis, there exis ts a resolutio n R ∗ for C over H B ,L 3 ( G 3 ), with each R p free with finite rank. Let C ∗ , ∗ be the first quadrant double complex given by C p,q = Hom bdd H B ,L 3 ( G 3 ) ( R p ˆ ⊗ T q , V ) ∼ = Hom bdd H B ,L 3 ( G 3 ) ( R p , Ho m bdd ( T q , V )) . Filter this complex by rows. F or a fixed q we have . . . δ R → C ∗− 1 ,q δ R → C ∗ ,q δ R → C ∗ +1 ,q δ R → . . . The b ounded contraction for the complex R ∗ induces a contraction on C ∗ ,q , so E p,q 1 = 0 for p ≥ 1 and E 0 ,q 1 = Hom bdd H B ,L 3 ( G 3 ) ( T q , V ) ∼ = Hom bdd H B ,L 2 ( G 2 ) ( P q , V ). The E 2 -term is pr ecisely B H ∗ ( G 2 ; V ), and the s pectral sequence collapses her e. Filter C ∗ , ∗ by columns. F or a fixed p we hav e . . . δ T → C p, ∗− 1 δ T → C p, ∗ δ T → C p, ∗ +1 δ T → . . . By adjointness, C p,q ∼ = Hom bdd ( R p , Ho m bdd ( T q , V )), where R p is finite dimen- sional with R p ∼ = H B ,L 3 ( G 3 ) ˆ ⊗ R p . Let d ∗ T : Hom bdd ( T q , V ) → Hom bdd ( T q +1 , V ) be the map induced by d T . It is c le ar that ker δ T = Hom bdd ( R p , ker d ∗ T ) and im δ T ⊂ Hom bdd ( R p , im d ∗ T ). That Ho m bdd ( R p , im d ∗ T ) ⊂ im δ T follows from finite dimensionality of R p . B -BOUNDED COHOMOLOGY AND APPLICA TIONS 39 Finite dimensionality also implies Hom bdd  R p , ker d ∗ T im d ∗ T  ∼ = Hom bdd ( R p , ker d ∗ T ) Hom bdd ( R p , im d ∗ T ) . Thu s this sp ectral sequence has E p,q 1 ∼ = Hom bdd H B ,L 3 ( G 3 ) ( R p , B H q ( H ; V )) and E p,q 2 ∼ = B H p ( G 3 ; B H q ( G 1 ; V )). By a sp ectral sequence compa rison, if ( G 1 , L 1 ) is V - B IC, so is ( G 2 , L 2 ). Conse- quently if ( G 1 , L 1 ) is B -SIC, iso cohomologic alit y holds for a ll H B ,L 2 ( G 2 )-mo dules V , implying ( G 2 , L 2 ) is B - SIC b y Theorem 3 .  5.2. The sp ectral sequence asso ciated to a group acting on a complex. F ollowing Section 1.6 of [ 40 ], supp ose that a finitely generated gr oup G acts co com- pactly on a n acyclic simplicial complex X w itho ut inv ersio n. F or a simplex σ of X , denote the s tabilizer of σ by G σ . Denote b y Σ a set of representativ es o f simplexes of X mo dulo the G a ction, and b y Σ q the q - dimensional r epresen tatives in Σ. Let C ∗ ( X ) denote the simplicial chain complex of X . As X is a cyclic, ther e is an exact sequence 0 ← C ← C 0 ( X ) ← C 1 ( X ) ← C 2 ( X ) ← . . . There is a direct-sum dec o mposition C q ( X ) ∼ = L σ ∈ Σ q C [ G/G σ ]. F or each σ ∈ Σ, let P σ k = C [ G × G σ ( G σ ) k +1 ], the usual s implicial s tr ucture on G σ induced up to a C [ G ]-mo dule. In this way , P σ • is a pro jective C [ G ] reso lution of C [ G/G σ ]. This yields a double complex (24) . . .   . . .   . . .   M σ ∈ Σ 0 P σ 2   M σ ∈ Σ 1 P σ 2 o o   M σ ∈ Σ 2 P σ 2 o o   . . . o o M σ ∈ Σ 0 P σ 1   M σ ∈ Σ 1 P σ 1 o o   M σ ∈ Σ 2 P σ 1 o o   . . . o o M σ ∈ Σ 0 P σ 0 M σ ∈ Σ 1 P σ 0 o o M σ ∈ Σ 2 P σ 0 o o . . . o o As each Σ q is finite, applying Ho m C [ G ] ( · , M ) yields the fo llowing double co mplex . 40 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY (25) . . . . . . . . . M σ ∈ Σ 0 Hom C [ G ] ( P σ 2 , M ) / / O O M σ ∈ Σ 1 Hom C [ G ] ( P σ 2 , M ) / / O O M σ ∈ Σ 2 Hom C [ G ] ( P σ 2 , M ) / / O O . . . M σ ∈ Σ 0 Hom C [ G ] ( P σ 1 , M ) / / O O M σ ∈ Σ 1 Hom C [ G ] ( P σ 1 , M ) / / O O M σ ∈ Σ 2 Hom C [ G ] ( P σ 1 , M ) / / O O . . . M σ ∈ Σ 0 Hom C [ G ] ( P σ 0 , M ) / / O O M σ ∈ Σ 1 Hom C [ G ] ( P σ 0 , M ) / / O O M σ ∈ Σ 2 Hom C [ G ] ( P σ 0 , M ) / / O O . . . Consider the sp ectral sequence arising from filtering this double complex by columns. That P σ ∗ be a pro jectiv e C [ G ] r esolution of C [ G/G σ ] mea ns that L σ ∈ Σ 0 P σ • is a pro jectiv e resolution of L σ ∈ Σ 0 C [ G/G σ ]. The E p,q 1 -term of this spectra l se- quence is then Ext q C [ G ]   M σ ∈ Σ p C [ G/G σ ] , M   ∼ = Y σ ∈ Σ p Ext q C [ G ] ( C [ G/G σ ] , M ) ∼ = Y σ ∈ Σ p Ext q C [ G σ ] ( C , M ) ∼ = Y σ ∈ Σ p H q ( G σ ; M ) On the o ther hand, the to ta l complex of the double co mplex in equation 24 , serves as a pr o jective resolution of C ov er C [ G ], yielding a theor em of Serre. Theorem 13 (Serre) . F or e ach C G -mo dule M , ther e is a sp e ctr al se quenc e with E p,q 1 ∼ = Q σ ∈ Σ p H q ( G σ ; M ) and which c onver ges to H p + q ( G ; M ) . This extends to the B -bounded cas e, when the stabilizers are given the length function restricted from G . Let C B m ( X ) b e defined as in the pr oof of Theorem 3 . If the higher weigh ted Dehn functions o f X are B -bo unded, C B ∗ ( X ) g ives a chain complex of co mplete b ornolog - ical H B ,L ( G )-modules , endo wed with a bounded C -linear cont racting ho motop y . There is a natural quotien t length, w , defined on G/G σ induced from the length L on G via w ( g G σ ) := min { L ( g h ) | h ∈ G σ } . Deno te by H B ,w ( G/G σ ) the completion C [ G/G σ ] under the following family of seminorms. | X x ∈ G/G σ α x x | λ := X x ∈ G/G σ | α x | λ ( w ( x )) λ ∈ B B -BOUNDED COHOMOLOGY AND APPLICA TIONS 41 There is a b ornologica l isomor phism C B q ( X ) ∼ = L σ ∈ Σ q H B ,w ( G/G σ ). Similarly , let B P σ k denote the corres ponding c ompletion of P σ k . As a bov e, we obtain a double complex, but of b ornologica l H B ,L ( G )-modules . (26) . . .   . . .   . . .   M σ ∈ Σ 0 B P σ 2   M σ ∈ Σ 1 B P σ 2 o o   M σ ∈ Σ 2 B P σ 2 o o   . . . o o M σ ∈ Σ 0 B P σ 1   M σ ∈ Σ 1 B P σ 1 o o   M σ ∈ Σ 2 B P σ 1 o o   . . . o o M σ ∈ Σ 0 B P σ 0 M σ ∈ Σ 1 B P σ 0 o o M σ ∈ Σ 2 B P σ 0 o o . . . o o F or a n y H B ,L ( G )-mo dule M , applying the b ounded e q uiv a rian t homomorphism functor Hom bdd H B ,L ( G ) ( · , M ) yields the following. (27) . . . . . . M σ ∈ Σ 0 Hom bdd H B ,L ( G ) ( B P σ 2 , M ) / / O O M σ ∈ Σ 1 Hom bdd H B ,L ( G ) ( B P σ 2 , M ) / / O O . . . M σ ∈ Σ 0 Hom bdd H B ,L ( G ) ( B P σ 1 , M ) / / O O M σ ∈ Σ 1 Hom bdd H B ,L ( G ) ( B P σ 1 , M ) / / O O . . . M σ ∈ Σ 0 Hom bdd H B ,L ( G ) ( B P σ 0 , M ) / / O O M σ ∈ Σ 1 Hom bdd H B ,L ( G ) ( B P σ 0 , M ) / / O O . . . As in the non-b ornolo gical case above, when filtering by columns we obtain a sp ectral sequence that conv erg es to the cohomo lo gy of the to ta l complex. The choice of w on G/G σ ensures a b ornolo g ical isomorphism E xt ∗ H B ,L ( G ) ( H B ,w ( G/G σ ) , M ) ∼ = E xt ∗ H B ,L ( G σ ) ( C , M ) . As above w e find E p,q 1 ∼ = Q σ ∈ Σ p B H q ( G σ ; M ). Moreov e r, the total co mplex of 26 gives a pro jectiv e res olution of C over H B ,L ( G ). This verifies the following theorem. 42 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Theorem 14. Supp ose al l higher weighte d Dehn funct io ns of C ∗ ( X ) ar e B -b ounde d, when the acycl ic c omplex X is e qu ipp e d with the 1 -skeleton weighting. F or e ach H B ,L ( G ) -mo dule M , ther e is a sp e ctr al se quenc e with E 1 -term the pr o duct of the B H ∗ ( G σ ; M ) which c onver ges to B H ∗ ( G ; M ) . By co mparison with the spectr al sequence fr o m Theorem 13 , we immediately obtain the following co r ollary . Corollary 6. Supp ose the acyclic c omplex X is e quipp e d with the 1 -skeleton weigh t- ing, and al l higher weighte d Dehn functions of C ∗ ( X ) ar e B - b ounde d. If M is a H B ,L ( G ) -mo dule for which e ach ( G σ , L ) is M - B IC, then ( G, L ) is M - B IC. In p ar- ticular if e ach ( G σ , L ) is B -SIC, so is ( G, L ) . 6. Duality groups and the comp arison map 6.1. Dualit y and P oincar ´ e Duali t y Groups. W e recall that G is a duality group of dimens io n n if there exists a G -mo dule D such that H i ( G, M ) ∼ = H n − i ( G, D ⊗ M ) If this is the ca s e, then D = H n ( G, Z [ G ]) is the dualizing module . When D = Z , the group is called a Poicar´ e Duality group . It is o rien ta ble precisely when the action of G on D (induced by the right a ction o f G on Z [ G ]) is trivial. All known orientable Poincar ´ e Duality gro ups oc c ur as the fundamental gr oup of a close d orientable aspherica l manifold. 6.2. Iso cohomolog icalit y and the fundamental class. The q ue s tion o f isoco- homologica lit y for oriented duality g roups is answered b y the following theorem. All homology a nd cohomolo gy groups are taken with co efficien ts in C . Theorem 15. L et M b e a c omp act, close d, orientable manifold of dim. n which is aspheric al ( f M ≃ ∗ ). L et G = π 1 ( M ) , and let µ ′′ G ∈ H n ( M × M ) denote the fundamental c ohomolo gy class in H ∗ ( M × M ) dual to t he diagonal emb e d- ding ∆( M ) ⊂ M × M 5 . If µ ′′ G is in the image of the c omp arison map Φ ∗ B : B H ∗ ( G × G ) → H ∗ ( G × G ) with r esp e ct to a length function L on G , then ( G, L ) is B -iso c ohomolo gic al. Pr o of. Ass ume L fixed, and co nsider the following diagra m: H i ( G ) − ∩ µ G / / H n − i ( G ) Φ B ∗   µ ′′ G / − o o B H i ( G ) Φ ∗ B O O − ∩ µ B G / / B H n − i ( G ) ? o o ❴ ❴ ❴ Here µ G ∈ H n ( G ) = H n ( M ) de no tes the fundamen tal homology c la ss of M . Now − ∩ µ G is an isomorphism with inv erse given by µ ′′ G / − . The homo logy class µ B G ∈ B H n ( G ) is simply the image of µ G ∈ H n ( G ) under the co mparison map Ψ B ∗ . By section 2.6 , − ∩ µ B G = Φ B ∗ ◦ ( − ∩ µ G ) ◦ Φ ∗ B In fact this iden tify follows from a simila r o ne that holds on the (co)chain level. If ther e exists a class B µ ′′ G ∈ B H n ( G × G ) s a tisfying µ ′′ G = Ψ ∗ B ( B µ ′′ G ), then taking 5 This class is s imply the image, under the restriction map H ∗ ( M × M , M × M − ∆( M )) → H ∗ ( M × M ), of the Thom class asso ciated to the normal bundle of the diagonal embedding. B -BOUNDED COHOMOLOGY AND APPLICA TIONS 43 ? = B µ ′′ G / − in the ab o ve diagr am and app ealing a g ain to section 5.1 , we g et the second iden tity µ ′′ G / − = Φ ∗ B ◦ ( B µ ′′ G / − ) ◦ Φ B ∗ This implies the dia gram, with “?” so defined, is commutativ e. The fact that the maps at the top are isomo rphisms then implies all o f the other maps in the diagram are a s well.  [Note: There is a different way of thinking a bout this result. By the Duality Theorem (Thm. 11 .10 of [ 29 ]), for any basis { b i } o f H ∗ ( G ) = H ∗ ( M ), taken as a (finite-dimensional) gr aded vector space over C , there exists a “dua l” ba sis { b ♯ j } with < b i ∪ b ♯ j , µ G > = δ ij . In terms of these bases, µ ′′ G is given by the equation µ ′′ G = X i ( − 1) dim ( b i ) b i × b ♯ i The condition that this cla s s is B -b ounded then forces each of the b i ’s (and hence also the b ♯ j ’s) to b e B -bounded, via linear indep endence.] When B G has the homo top y t yp e of a n or ien ted manifold with b oundary , w e hav e a similar result. Theorem 16. Su pp ose ( G, L ) is a gr oup with wor d-length, su ch that B G ≃ M an oriente d c omp act n -dimensio nal manifol d with c onne cte d b oundary ∂ M . Assu me also that ∂ M is aspheric al, and inc ompr essibly emb e dde d in M (i.e., the induc e d map on fun damental gr oups π 1 ( ∂ M ) → π 1 ( M ) is inje ct ive). L et D ( M ) = M ∪ ∂ ( M ) M denote the double of M along its b oundary. If the fundamental c ohomolo gy classes of b oth D ( M ) and ∂ M ar e b oth in the image of the c omp arison m ap for a b ounding class B (in the manner describ e d by the or em 15 ), then G is B - iso c ohomolo gic al. Pr o of. Le t G ′ i = π 1 ( ∂ M ) and G ′′ = π 1 ( D ( M )). By V an Kamp en’s theorem, G ′′ ∼ = G ∗ G ′ G ; moreover, the incompressibility of ∂ M in M implies D ( M ) ≃ K ( G ′′ , 1 ) is aspherical. Now c o nsider the diagram . . . / / B H j − 1 ( G ′ ) δ / /   B H j ( G ′′ ) / /   B H j ( G ) ⊕ B H j ( G ) / /   B H j ( G ′ ) / /   . . . . . . / / H j − 1 ( G ′ ) δ / / H j ( G ′′ ) / / H j ( G ) ⊕ H j ( G ) / / H j ( G ′ ) δ / / . . . Both the top and bo ttom sequences are derived from the collapsing of the spec- tral sequence asso ciated to a g roup acting o n a co mplex (in this case, a tr ee with t wo edges and three vertices, representing the amalgamated free pro duct). The vertical ma ps are induced by the compariso n tra nsformation B H ∗ ( − ) → H ∗ ( − ), implying the diagram is commutativ e. By Theorem 15 , the compariso n map is an isomorphism for b oth G ′ and G ′′ (bo th of whose clas sifying spaces are r epresen ted by compact, oriented finite-dimensional manifolds without b oundary). The result follows b y the five-lemma.  It is not clear if this is the b est p ossible result when the b oundary is non-empty , i.e., whether B -iso cohomolog ic a lit y for G could b e guaranteed b y the B -b oundedness of a sing le cohomolo g y class. It is als o not clear what one can say in g eneral if either ∂ M is not aspherica l, o r if it is, but not incompressibly embedded in M . All of these situations would seem to deserve further atten tion. 44 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY 6.3. B -duality groups. Using the pairing op e rations of section 2.6 , one has an obvious extension of the definition of a duality group to the B -bo unded setting. Definition 16. Given a b ounding class B and a gr oup with wor d-length ( G, L ) , we say that G is a B -duality gr oup of dimension n if ther e exists an H B ,L ( G ) -mo dule D B and a “fundamental class” µ B ∈ B H n ( G, D B ) with B H i ( G, V ) − ∩ µ B − → ∼ = B H n − i ( G, D b ⊗ V ) for al l H B ,L ( G ) mo dules V . Theorem 17. L et B b e a b ounding class, and ( G, L ) a B -duality gr oup with duality mo dule D B . Supp ose µ B is in t he image of the c omp arison map Ψ B ∗ . Then • If D B is finite-dimensional over C , ( G, L ) is str ongly mono c ohomolo gic al (that is, the c omp arison map is inje ctive in c ohomolo gy for al l b ornolo gic al H B ,L ( G ) -mo dules V . • If D B is infinite-dimensional over C , ( G, L ) is mono c ohomolo gic al for al l b ornolo gic al H B ,L ( G ) -mo dules V whi ch ar e fin ite-dimensio nal over C . Pr o of. Cho ose µ D ∈ H n ( G, D ) with Φ B n ( µ D ) = µ B D . W e ca n cons ider a diagr am analogo us to that of Theorem ( 15 ): H i ( G, V ) − ∩ µ D / / H n − i ( G, D B ⊗ V ) Φ B ∗   B H i ( G, V ) Φ ∗ B O O − ∩ µ B D / / B H n − i ( G, D B b ⊗ V ) A t issue in this dia gram is the difference b et w een D B ⊗ V and D B b ⊗ V . How ever, if either D B or V is finite-dimensiona l ov er C , this difference v anishes a nd the dia- gram comm utes, verifying injectivit y of the co mparison map in the ca ses indicated. Note that we do not assume the top horizo n tal map in the ab o ve diagram is an isomorphism.  Remark Ideally , o ne would like to pro ve the diagr am commutes whenever µ B is in the ima g e o f the comparis on map. How ever, we hav e not y et been able to show this. 6.4. Tw o solvmanifol ds. W e construct examples o f groups π , admitting closed oriented compact manifold mo dels for B π of small dimension, for which the com- parison map fails to be surjective. Let L n denote the standard word-length function on Z n , a nd set LW L n ( g ) = lo g(1 + L n ( g )) This is still a length function on Z n , but it is not B -eq uiv a len t to L n unless E  B . Prop osition 3 . L et B H ∗ log ( Z n ) denote t he B -b ou n de d c ohomolo gy of the gr oup with wor d-length ( Z n , L W L n ) . Then for al l B ≺ E , 0 = Φ ∗ B : B H ∗ log ( Z n ) → H ∗ ( Z n ) , ∗ > 0 Pr o of. When ∗ = 1, ele ments o f B H 1 log ( Z m ) corresp ond bijectiv ely to gro up homo- morphisms from Z m to C , equipp ed with its usua l no rm. The norm of an y non-zero homomorphism grows linear ly w ith res pect to the standard word-length function on Z . This means it grows exp onen tially as a function of LW L m . When B ≺ E , this B -BOUNDED COHOMOLOGY AND APPLICA TIONS 45 is impossible, implying B H 1 log ( Z m ) = 0 for all m ≥ 1. T his verifies the pro position in the case n = 1. Suppo se now that n > 1. There is a commut ing diagra m o f short-exact sequences of g roups with word-length Z n − 1 LW L n − 1 / /   Z n − 1 LW L n − 1 × Z st / /   Z st   Z n − 1 st / / Z n st / / Z st By induction, w e may a ssume the co mparison map B H ∗ log ( Z n − 1 ) → H ∗ ( Z n − 1 ) is zero for ∗ > 0. Both s equences s atisfy the conditions for the Ser r e sp ectral sequence in B -b ounded cohomology to exist. A sp ectral sequence argument then shows the vertical map in the middle Z n − 1 LW L n − 1 × Z st → Z n st m ust be z ero in B -b ounded cohomolog y for ∗ > 1 , implying the same fo r the comp osite map Z n LW L n → Z n − 1 LW L n − 1 × Z LW L 1 → Z n − 1 LW L n − 1 × Z st → Z n st Moreov er, for the standard word-length the compar ison map induces an isomo r- phism B H ∗ st ( Z n ) ∼ = → H ∗ ( Z n ). W e may then co nclude that the compar ison ma p B H ∗ log ( Z n ) → H ∗ ( Z n ) is zero for ∗ > 1. As w e hav e a lr eady s ho wn it is zero for ∗ = 1 , this completes the induction step.  Example 1 [Gromov] As ab ov e, as s ume L  B ≺ E and let G b e the semi-direct pro duct Z 2 ⋊ Z , where Z act on Z 2 by the representation  2 1 1 1  This is a split-extens ion of Z by Z 2 ; moreover, Z 2 has exp onential distortion in G . This is equiv alent to saying the induced w o rd-length function on Z 2 coming from the embedding in G is (linear ly ) equiv alen t to LW L 2 . F o r the base gr oup Z , the word-length function induced by the pro jection G ։ Z is the standard one. Now the Hochschild-Serre sp ectral sequence in ordinar y cohomolog y for this extension satisfies E ∗∗ 2 = E ∗∗ ∞ for dimensiona l rea sons. E m b edding Z 2 ⋊ Z in the solv able Lie gro up R 2 ⋊ R , the action of the base on the fib er (ov er R ) is simila r to the a ction given b y r ◦ ( r 1 , r 2 ) = ( e λr r 1 , e − λr r 2 ). The first exterio r p ow er of this representation has no inv ariant subspaces, while the s econd exter ior power is the ide ntit y . Hence E 0 , 1 2 = H 0 ( Z ; H 1 ( Z 2 )) = 0, while E 0 , 2 2 = H 0 ( Z ; H 2 ( Z 2 )) ∼ = C . On cla ssifying spaces the short- e xact sequence Z 2 ֌ G ։ Z pr o duces a fibra tion sequence of closed oriented manifolds a nd or ien tatio n-preserving maps. This yields a Poincar´ e Dualit y map on the E ∗∗ 2 -term of the sp ectral sequence for H ∗ ( G ). By this duality , w e conclude E 0 , 1 2 is dual to E 1 , 1 2 which there fore m ust also b e zero (w e alrea dy knew H 1 ( Z ) = E 1 , 0 2 ∼ = E 0 , 2 2 = H 0 ( Z ; H 2 ( Z 2 )) ∼ = C ). This gives an isomorphism H ∗ ( G ) ∼ = H ∗ ( Z ) ⊗ H ∗ ( Z 2 ), although there is no homomor phism of groups inducing it. If we denote by t i ∈ H i ( G ) the element corr esponding to the generator of H i ( Z i ) , i = 1 , 2 (after fixing a prefer red or ien tation of B G ), then Prop osition 4. The c ohomolo gy class t 2 ∈ H 2 ( G ) ∼ = C c annot lie in the image of the c omp arison map Ψ 2 B : B H 2 ( G ) → H 2 ( G ) whenever B ≺ E . 46 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY Pr o of. The compar ison map is natural with r espect to those maps induced by gr oup homomorphisms, implying the existence of a comm uting diagram B H 2 ( G ) / / Ψ 2 B   B H 2 log ( Z 2 ) Ψ 2 B   H 2 ( G ) / / / / H 2 ( Z 2 ) F or B ≺ E , the map on the right is trivial by Prop osition 3 , while the s pectral sequence argument for ordinary cohomology just given shows the low er horizontal map sends t 2 non-trivially to the g enerator of H 2 ( Z 2 ) ∼ = C . Thus t 2 cannot b e in the image o f Ψ 2 B (this is in the spir it to Gromov’s original argument referenced ab o ve).  With so me additional work, one ca n als o s ho w t 1 t 2 ∈ H 3 ( G ) is not in the image of Ψ 3 B whenever B ≺ E . Of cours e, by Theorem 15 , The dual fundamen tal class u ′ ∈ H 3 ( G × G ) cannot be in the ima ge of Ψ 3 B for B ≺ E . There are some additional consequences o f this first example worth noting (with P  B ≺ E ). • All surface g roups are non-p ositively curv ed - hence B -IC - so 3 is the low est dimension for which there can exis t a closed o rien ted K ( π , 1) manifold with non- B -b ounded co homology . • Nilp oten t groups are B -IC when P ≺ B [ 34 ], [ 2 3 ] , so solv a ble groups are the simples t types of groups which could have non- B - bounded cohomo logy for P  B . • F o r finitely-genera ted gr oups, all 1-dimensio nal co homology classes exhibit linear growth with respe c t to the word-length function, so cohomological dimension 2 is the first dimensio n in whic h class es not B -bounded with resp ect to the w ord-length function co uld o ccur. • If the fir st Dehn function of G were B -bo unded, G would ha ve to be strong ly B -iso cohomologica l in cohomology dimens io ns 1 and 2. By con tradiction, we r eco ver the result o f Gersten [Ge1] that the first Dehn function of G m ust be (at least) exponential. Example 2 [Arzhantsev a-Osin] Let φ : Z 2 → S L 3 ( Z ) b e an injectio n sending the usual genera tors o f Z 2 to to semi-simple ma trices with real sp ectrum. Denote by H b e the semi-dire c t pro duct Z 3 ⋊ Z 2 where Z 2 acts via the repr esen tation induced b y φ . The classifying space B H is homotopy-equiv alent to a 5-dimensional closed, com- pact, and or ien ted solvma nifold M 5 . It is shown in [ 3 ] that Z 3 is exp onent ially distorted in H in a manner s imila r to the previous example. Theorem 18. Ther e exists a c ohomo lo gy class t 3 ∈ H 3 ( H ) not in the image of Ψ 3 B for any B ≺ E . Pr o of. O n the level of c lassifying spaces, the short-exact sequence Z 3 i ֌ H p ։ Z 2 corres p onds to a fibration sequence of closed oriented compact manifolds, with the maps pr eserving o rien tation. Thus the top-dimensiona l cohomolog y class µ 5 ∈ B -BOUNDED COHOMOLOGY AND APPLICA TIONS 47 H 5 ( H ) sa tis fie s µ 5 = µ 3 µ 2 where µ 3 maps under i ∗ to 0 6 = µ ′ 3 ∈ H 3 ( Z 3 ) Z 2 (the Z 2 - inv ar ian t fundamen tal cohomo logy class of Z 3 ), a nd µ 2 = p ∗ ( µ ′ 2 ) where µ ′ 2 ∈ H 2 ( Z 2 ) is the fundamental cohomology class for Z 2 . As before , there is a commuting dia gram B H 3 ( H ) / / Ψ 3 B   B H 3 log ( Z 3 ) Ψ 3 B   H 3 ( G ) / / / / H 3 ( Z 3 ) where the map on the righ t is zero. The result follows.  By a more detailed a nalysis, o ne ca n conclude that µ 5 ∈ H 5 ( H ) ∼ = C is not B -b ounded for a n y B ≺ E , and by The o rem 15 , w e kno w the same for the dual fundamen tal class in H 5 ( H × H ). Ho wev er, this ex ample is imp ortan t for another reason. Corollary 7. The Dehn functions of H ar e not B -e quivalent for any b ounding class B ≺ E . Pr e cisely, the first Dehn function is quadr atic, while the se c ond Dehn function is at le ast simple exp onential. Pr o of. The fir s t Dehn function of H was co mputed in [ 3 ], where it was shown to be qua dratic. If the seco nd Dehn function w er e B -b ounded for some L  B ≺ E , then by Theore m 2 , the group H would ha ve to b e B -iso cohomolog ical through dimension 3 contradicting the pr evious result. So the s e cond Dehn function must be at least simple exp onential.  6.5. More on the comparison map. W e hav e shown the comparison map fails to b e surjective in gener al, a t least for bounding classes B ≺ E . It is natural to ask whether this map also fails to b e injective. The next theorem answers this questio n. Theorem 19. L et ( G, L st ) b e a discr ete gr oup with standar d wor d-length fun ction, with B G ≃ Y a fin ite c omplex. If B is a b ou n ding class for which t he c omp arison map Φ ∗ B ( G ) : B H ∗ ( G ) → H ∗ ( G ) fails to b e surje ctive, t hen ther e is another gr oup F ( G ) , dep ending fun ct oria l ly on G u p to homotopy, for which t he c omp arison map Φ ∗ B fails to b e inje ctive. Pr o of. As B G is ho motopically finite, w e ma y construct a finitely- generated hyper- bo lic group H ( G ) and a map p G : H ( G ) → G which induces an injection in gro up cohomolog y with tr ivial co efficients [ 18 , 12 , 1 4 ]. Also , for a ny dis crete gr oup G ′ , a classical construction allow us to embed G ′ in an a c yclic group A ( G ′ ), wher e the inclusion i G ′ : G ′ ֒ → A ( G ′ ) is a functor ial constructio n in G ′ . If G ′ is finitely- generated and equipp ed with the standar d word-length function, we ca n arra nge for the imag e of G ′ in A ( G ′ ) to b e non- distorted. Abbr e v iate H ( G ) as C , and let A 1 = G × A ( C ), A 2 = A ( C ). There are inclusions C ֒ → A 1 , g 7→ ( p G ( g ) , i C ( g )) , (28) C ֒ → A 2 , g 7→ i C ( g ) (29) Let A 3 = A 1 ∗ C A 2 . By the spe c tral sequence of sec tion 5.2 , there is a commuting diagram of May er-Vietoris s e q uences 48 R ONGHUI JI, CRICHTON OGLE, AND BOBBY W. RAMSEY . . . / / B H j − 1 ( C ) δ / / ∼ =   B H j ( A 3 ) / /   B H j ( A 1 ) ⊕ B H j ( A 2 ) / /   B H j ( C ) / / ∼ =   . . . . . . / / H j − 1 ( C ) δ / / H j ( A 3 ) / / H j ( A 1 ) ⊕ H j ( A 2 ) / / H j ( C ) δ / / . . . Because C is finitely-generated hyperb olic, the co mpa rison map fo r C is a n isomorphism for all L  B . Mor eo ver, H ∗ ( A 2 ) = 0 for ∗ > 0, a nd F ∗ ( A 1 ) ∼ = F ∗ ( G ) ⊗ F ∗ ( A ( C )) fo r F ∗ ( − ) = B H ∗ ( − ) , H ∗ ( − ). Hence the cokernel of the compar ison map for A 1 is naturally iso morphic to the co k er nel of the compar ison map for G . The injectivit y of H ∗ ( G ) → H ∗ ( C ) implies the map H j ( A 3 ) → H j ( A 1 ) ⊕ H j ( A 2 ) is zero for j > 0. The result is a n injection coker (Φ ∗ B : B H ∗ ( A 1 ) ⊕ B H ∗ ( A 2 ) → H ∗ ( A 1 ) ⊕ H ∗ ( A 2 )) ֒ → ker  Φ ∗ +1 B : B H ∗ +1 ( A 3 ) → H ∗ +1 ( A 3 )  Define F ( G ) = A 3 . If co k er(Φ m B ( G )) 6 = 0, then k er(Φ m +1 B ( F ( G ))) 6 = 0. The acyclic gr oup construction G 7→ A ( G ) can b e done functorially , as can the hyper- bo lization of the finite co mplex Y . Ho wev er, this re q uires cho osing a finite complex Y ≃ B G , whic h, o n the categ ory of t yp e finitely- presen ted F L gro ups, is functorial only up to homotopy .  Corollary 8. Ther e exist discr ete gr oups e quipp e d with standar d wor d-length fu n c- tion for which the c omp arison map Φ 3 B fails to b e inje ctive for al l B ≺ E . Pr o of. Le t G b e the gr o up in Prop osition 4 . By the prev io us theorem, Φ 3 B ( F ( G )) cannot be an injection for any B ≺ E .  It should b e noted that the g roups res ulting fro m the ab ov e constructio ns will t ypically hav e large classifying spaces, even when B G ha s the ho motop y t yp e of a relatively simple complex. The following alternativ e construction provides a more geometric mo del for the acyclic “envelope” used ab ov e. Again, assume G is type F L , so that B G ≃ Y a finite co mplex. Accor ding to recent w o r k of Leary [L], we may construct a dia gram T Y / / / /   T b Y   Y / / / / b Y where b Y denotes the cone on Y (whic h can b e done so as to b e functorial in Y and preser ve finiteness), and wher e T X denotes the “metric” K an-Th urston space ov er X . By [L ], this is a CA T(0)-spa ce (hence aspherical) whose co ns truction is functoria l on the categ ory of finite complexes , for which the map T X → X is a homology isomorphism. Thus in the above setup, we can replace C by C 1 := π 1 ( T Y ) and A ( C ) by C 2 := π 1 ( b Y ), and rep eat the construction with A 1 = G × C 2 , A 2 = C 2 , F ( G ) = A 3 = A 1 ∗ C 1 A 2 , the difference now being that A 1 , A 2 as well as the amalgama ted pro duct A 3 are all of type F L . Because CA T(0 )- groups admit a synchronous linear com bing, they ar e B -SIC for all B  P . Hence Theorem 20 . L et ( G, L st ) b e a gr oup of t yp e F L with standar d wor d-length func- tion, wher e B G ≃ Y a finite c omplex. If B  P is a b ounding class for which t he B -BOUNDED COHOMOLOGY AND APPLICA TIONS 49 c omp arison map Ψ B ( G ) : B H ∗ ( G ) → H ∗ ( G ) fails to b e surje ctive, t hen t her e is another gr oup F ( G ) of t yp e F L , dep ending functorial ly on G up to homotopy , for which the c omp arison map Φ ∗ B fails to b e inje ctive. Corollary 9. 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Blackf ord St., Indianapolis, IN 46202 USA E-mail addr ess : ronji@mat h.iupui.e du Dep ar tmen t of Ma thematic s,The Ohio S t a te University, 231 W. 18 th A ve., Columbus, OH 43210 USA E-mail addr ess : ogle@math .ohio-sta te.edu Dep ar tmen t of Ma thematic s,The Ohio S t a te University, 231 W. 18 th A ve., Columbus, OH 43210 USA E-mail addr ess : ramsey.31 3@math.os u.edu