The limit of F_p-Betti numbers of a tower of finite covers with amenable fundamental groups

THE LIMIT OF F p -BETTI NUMBERS OF A TO WER OF FINITE CO VERS WITH AME NABLE FUNDAMENT AL GR OUPS PETER LINNELL, WOLF GANG L ¨ UCK, AND ROMAN SA UER Abstract. W e prov e an analogue of the Approx imation Theorem of L 2 -Betti num b ers b y Betti n umbers for arbitrary coefficient fields a nd virtually torsionfree amenable groups. The limit of Betti num b ers is i den tified as the di mension of some mo dule ov er the Ore lo cali zation of the group ring. 0. I ntroduction A r esidual chain o f a group G is a sequence G = G 0 ⊃ G 1 ⊃ G 2 ⊃ · · · of normal subgroups of finite index such that T i ≥ 0 G i = { e } . The n -th L 2 -Betti num ber of any finite free G -CW complex X is the limit of the n -th Betti nu mbers of G i \ X normalized by the index [ G : G i ] for i → ∞ [11]. If we instead consider Betti num be rs b n ( G i \ X ; k ) with r esp ect to a field of characteristic p > 0, the questions whe ther the limit exists, what it is, and whether it is independent of the r esidual chain ar e co mpletely op en for arbitra ry re s idually finite G . F or G = Z k and every field k E lek show ed that lim i →∞ b n ( G i \ X ; k ) exists a nd expre sses it in terms of the ent ro py o f G -a ctions on the P ontrjagin duals of finitely genera ted k G - mo dules [4] – his tec hniques play a n impor tant role in this pap er (see Section 1.3). It was observed in [1, Theor em 17] that the mere con vergence of the right hand side of (i) in Theore m 0.2 for every amenable G and every field k follows from a general co nv ergence principle fo r subadditive functions on a menable g roups [9 ] and a theorem by W e is s [18]. The main purpo se of this pap er is to determine the limit lim i →∞ b n ( G i \ X ; k ) in algebraic terms for a large class of amenable groups including virtual torsionfree elemen tary amenable groups. This makes the limit computable by homo logical techniques; se e e.g ., the sp ectral sequence argument o f E x ample 6 .3. More precisely , the limit will be expressed in terms of the Or e dimension . The group ring k G of a tor sionfree amenable group satisfying the zer o-diviso r conjecture fulfills the Ore co nditio n with r esp ect to the subset S = k G − { 0 } [1 2, Example 8.16 o n page 324 ]; we will review the Ore lo caliza tion in Subse ction 1.1. The O r e lo calization S − 1 k G is a skew field containing k G . Therefore the following definition makes sense: Definition 0.1 (Ore dimens io n) . Let G be a to r sionfree amenable group such that kG contains no z e ro-diviso rs. The Or e dimension of a k G -mo dule M is defined by dim Ore kG ( M ) = dim S − 1 kG  S − 1 k G ⊗ kG M  . Date : March 1, 2010. 2000 Mathematics Subje ct Classific ation. 16U20, 55P99. Key wor ds and phr ases. Amenabilit y , Ore lo calization, Betti n umbers. The authors thank the HIM at Bonn for its hospitality during the T r imester program “Rigidity” in the fall 2009 when this paper was written. This work was financially supported by the Leibniz-Preis of the second author. 1 2 PETER LINNELL, W OLFGANG L ¨ UCK, AND ROMAN SA UER The following theor em is our main result; we will prove a mor e genera l version, including virtually torsionfree groups, in Sectio n 5. Theorem 0.2. L et k b e a field. L et G b e a torsionfr e e amenable gr oup for which kG has no zer o-divisors 1 . L et ( G n ) n ≥ 0 b e a r esidual chain of G . Then: (i) Consider a finitely pr esente d k G -mo dule M . Then dim Ore kG ( M ) = lim n →∞ dim k  k ⊗ kG n M  [ G : G n ] ; (ii) Consider a finite fr e e k G -chain c omplex C ∗ . Then we get for al l i ≥ 0 dim Ore kG  H i ( C ∗ )  = lim n →∞ dim k  H i ( k ⊗ kG n C ∗ )  [ G : G n ] ; (iii) L et X b e a finite fr e e G - C W -c omplex. Then we get for al l i ≥ 0 dim Ore kG  H i ( X )  = lim n →∞ dim k  H i ( G n \ X ; k )  [ G : G n ] . Remark 0.3 (Fields of characteristic zero) . Let G b e a group with a r esidual chain ( G n ) n ≥ 0 , and let M be a finitely pre sented k G -mo dule. Then the Approximation Theo rem for L 2 - Betti n umbers says that (0.4) dim N ( G )  N ( G ) ⊗ kG M  = lim n →∞ dim k  k ⊗ kG n M  [ G : G n ] provided k is an algebraic num ber field. Here N ( G ) is the group von Neumann alg ebra, and dim N ( G ) is the von Neumann dimension. See [11] for k = Q and [3] for the gener al case . Let k b e a field of characteristic zero and let u = P g ∈ G x g · g ∈ k G b e an element. Let F be the finitely ge ne r ated field extension of Q given by F = Q ( x g | g ∈ G ) ⊂ k . Then u is already an element in F G . The fie ld F embeds into C : since F is finitely generated, it is a finite algebraic extension of a transcendental extension F ′ of Q [8, Theorem 1.1 on p. 356], and F ′ has finite transcendence degree o ver Q . Since the transcendence degr ee o f C ov er Q is infinite, there exists an embedding F ′ ֒ → C induced by an injection of a trans c e ndence basis of F ′ / Q in to a transcendence basis C / Q , whic h extends to F ֒ → C bec ause C is a lgebraica lly closed. This reduces the case of fields of characteristic zer o to the ca se k = C . In [6] Elek prov ed (0.4) for amenable G a nd k = C (see also [13]). Moreov er, if G is a to rsionfree amenable gro up such that C G contains no zero-divisors and k is a field of characteristic zer o , then dim N ( G )  N ( G ) ⊗ kG M  = dim Ore kG ( M ) . This follows from [12, Theorem 6.3 7 on pag e 259, Theo rem 8.29 o n page 3 30, Lemma 1 0.16 on pa ge 376, a nd L e mma 1 0.39 o n page 388]. In particula r, Theore m 0.2 follows for k of characteristic z ero. So the interesting new case is the one of a field of prime character is tic. 1 This assumption is satisfied if G is torsionfree elementary amenable. See Remark 1.1 THE LIMIT OF BETTI NUMBERS OF A TO WER OF FINITE COVERS 3 1. Review of Ore localiza tion and E lek’s dimension function 1.1. Ore l o calization. W e review the Ore lo calization o f rings. F or pro ofs and more in- formation the reader is referred to [17]. Consider a tors ionfree g roup G and a field k . Let S b e the set of no n-zero-div isors o f k G . This is a multiplicativ ely clos e d subset o f k G a nd contains the unit element of k G . Suppose that k G satisfies the Kapla nsky Conje cture or zer o-divi sor c onje ctu r e , i.e., S = k G − { 0 } . F urther assume that S satisfies the left Or e c onditio n , i.e., for r ∈ k G and s ∈ S there exis ts r ′ ∈ k G and s ′ ∈ S with s ′ r = r ′ s . The n we ca n consider the Or e lo c alization S − 1 k G . Recall that every ele ment in S − 1 k G is of the form s − 1 · r for r ∈ k G and s ∈ S and s − 1 0 · r 0 = s − 1 1 · r 1 holds if and only there exists u 0 , u 1 ∈ R satis fying u 0 r 0 = u 1 r 1 and u 0 s 0 = u 1 s 1 . Addition is given on r epresentativ es by s − 1 0 r 0 + s − 1 1 r 1 = t − 1 ( c 0 r 0 + c 1 r 1 ) for t = c 0 s 0 = c 1 s 1 . Multiplication is given on repre s en- tatives by s − 1 0 r 0 · s − 1 1 r 1 = ( ts 0 ) − 1 cr 1 , where cs 1 = tr 0 . The zer o element is e − 1 · 0 and the unit element is e − 1 · e . The Or e lo caliza tion S − 1 k G is a skew field and the ca nonical map k G → S − 1 k G sending r to e − 1 · r is injective. The functor S − 1 k G ⊗ kG − is exact. Remark 1.1 (The Or e conditio n for gr oup ring s) . If a torsio nfree amena ble group G satisfies the Kaplansky Conjecture, i.e., k G contains no zer o-divisor , then for S = k G − { 0 } the Ore lo calizatio n S − 1 k G exists and is a s kew field [12, Example 8.16 on page 324]. Every torsionfree elemen tary amenable group satisfies the as sumptions above for all fields k [7 , Theorem 1.2 ; 10, Theorem 2.3 ]. If the group G contains the free group of rank tw o a s subgroup, then the Or e co ndition is never satisfied for k G [10, Prop o s ition 2.2]. F rom the pre v ious remark a nd the discussio n ab ove we o btain: Theorem 1.2. L et G b e a torsionfr e e amenable gr oup s u ch t hat k G c ontains no zer o- divisors. Then the Or e dimension dim Ore kG has the fol lowing pr op erties: (i) dim Ore kG ( k G ) = 1 ; (ii) F or any s hort exact se quenc e of k G -mo dules 0 → M 0 → M 1 → M 2 → 0 we get dim Ore kG ( M 1 ) = dim Ore kG ( M 0 ) + dim Ore kG ( M 2 ) . 1.2. Cross e d pro ducts, Goldie rings, and the generalized Ore lo calization. Throug h- out, let G b e a group, let k b e a s kew field. Let R b e a ring. The notion of crosse d pro duct gener alizes the one of group ring. A cr osse d pr o duct R ∗ G = R ∗ c,τ G is determined by maps c : G → aut( R ) and τ : G × G → R × such that, roughly sp eaking, c is a homomor phism up to the 2-co cycle τ . W e refer to the survey [12, 10.3.2 o n p. 398] for deta ils. If G is an extension of H b y Q , then the gro up ring k G is iso morphic to a crosse d pro duct k H ∗ Q . Some results in this pap er a re formulated for crossed pro ducts, although we only need the ca se of gr oup r ings for Theor e m 0.2. So the reader may think of gr oup rings most o f the time. How ever, crosse d pro ducts show up naturally , e.g., in pr oving that the vir tual O r e dimension (5.1) is well defined. W e r ecall the following definition. Definition 1 . 3. A ring R is left Goldie if there exists d ∈ N s uch tha t every dir ect sum of nonzero left ideals of R has at most d summands a nd the left a nnihilators a ( x ) = { r ∈ R ; rx = 0 } , x ∈ R , satisfy the maximum co ndition for ascending chains. A r ing R is prime if fo r a ny tw o ideals A, B in R , AB = 0 implies A = 0 or B = 0. 4 PETER LINNELL, W OLFGANG L ¨ UCK, AND ROMAN SA UER The subgroup of G genera ted by its finite normal subgr o ups will b e denoted by ∆ + ( G ). Then ∆ + ( G ) is also the set of elements of finite order which have only finitely many conju- gates. W e need the following thre e re sults: Lemma 1.4 ([15, Corollar y 5 of Lec tur e 4]) . If ∆ + ( G ) = 1 , then k ∗ G is prime. Theorem 1.5 ([14, Theor e m 4.1 0 on p. 456]) . The s et of n on-zer o-diviso rs in a prime left Goldie ring satisfies the Or e c ondition. The Or e lo c ali zation S − 1 R is isomorphic t o M d ( D ) for some d ∈ N and some skew field D . Theorem 1.6. If G is amenable and k ∗ G is a domain, then k ∗ G is a prime left Goldie ring. If G is an elementary amenable gr oup such that the or ders of the fi nite sub gr oups ar e b ounde d, then k ∗ G is left Goldie. Pr o of. If G is amenable a nd k ∗ G is a do main, then k ∗ G s a tisfies the Ore condition [3, Theorem 6.3], thus its Ore loc alization with respec t to S = k ∗ G − { 0 } is a skew field. By [14 , The o rem 4 .10 o n p. 45 6] k ∗ G is a prime left Goldie ring. The s econd assertio n is taken from [7 , Pr op osition 4 .2].  Next we extend the definition o f Ore dimension to pr ime left Goldie rings. Let R b e such a r ing. The functor S − 1 R ⊗ R − will still b e exact [17, Prop os itio n I I.1.4 on pa ge 51]. If M is a left R - mo dule, then S − 1 R ⊗ R M will b e a dir ect sum of n ir reducible S − 1 R -mo dules for some non-negative in teger n , and then the (gener alize d) Or e dimension of M is defined as dim Ore R ( M ) = n d . Since S − 1 R ∼ = M d ( D ) (Theorem 1.5) and M d ( D ) decomp oses into d copies of the irre ducible mo dule D d , we hav e dim Ore R ( R ) = 1. 1.3. Elek’s dimensio n function. Thr oughout this s ubsection le t G b e a finitely generated amenable group. W e revie w Ele k ’s de finitio n [5] o f a dimension function dim Elek kG for finitely generated k G -mo dules. Fix a finite set o f gener ators and equip G with the asso cia ted word metric d G . A Følner se quenc e ( F n ) n ≥ 0 is a sequence of finite subsets of G such that for a ny fixed R > 0 we have lim n →∞ | ∂ R F n | | F n | = 0 , where ∂ R F n = { g ∈ G | d ( g , F k ) ≤ R and d ( g , G \ F k ) ≤ R } . Let k b e an ar bitrary skew field endowed with the discrete top olog y and let N deno te the p ositive integers { 1 , 2 , . . . } . Let n ∈ N . W e equip the space of functions map( G, k n ) = Q g ∈ G k n with the pro duct top olog y , which is the same a s the top olog y of p oint wise co nv er- gence. The natural right G -action on ma p( G, k n ) is defined b y ( φg )( x ) = φ ( xg − 1 ) for g , x ∈ G, φ ∈ map( G, k n ). Also map( G, k n ) is a right k -vector space b y defining ( φk )( x ) = φ ( x ) k . F or a ny subset S ⊂ G and any subset W ⊂ map( G, k n ) let W | S = { f : S → k n | ∃ g ∈ W with g | S = f } . A right k -linear subspace V ⊂ map( G, k n ) is ca lled invariant if V is close d and inv aria nt under the rig ht G -a ction. THE LIMIT OF BETTI NUMBERS OF A TO WER OF FINITE COVERS 5 Elek defines the aver age dimension dim A G ( V ) of a n inv aria nt subspace V by choosing a Følner sequence ( F n ) n ∈ N of G and setting (1.7) dim A G ( V ) = lim s up n →∞ dim k  V | F n  | F n | . Theorem 1.8 ([5, Prop. 7 .2 and Prop. 9.2]) . The se quenc e in (1.7) c onver ges and its limit dim A G ( V ) is indep endent of the choic e of t he Følner se quenc e. Remark 1.9. Elek actually defines dim A G ( V ) using Følner exha ustions, i.e. incr e asing Følner sequences ( F n ∈ N ) with S n ∈ N F n = G . This makes no differ ence s ince the ex istence of the limit of (dim k  V | F n  / | F n | ) n ∈ N for arbitr ary Følner seque nc e s (and th us its indep endence of the c hoice) follows from [9, Theorem 6.1]. Let M b e a finitely gener a ted left k G -mo dule. The k -dual M ∗ = hom k ( M , k ) (where M and k a re viewed a s left k - mo dules, a nd ( φa ) m = φ ( am ) for φ ∈ M ∗ , a ∈ k and m ∈ M ) carries the natural right G -a ction ( φg )( m ) = φ ( g m ). The dual o f the free left k G -mo dule k G n is ca nonically isomor phic to map( G, k n ). An y left k G -surjectio n f : k G n ։ M induces a r ight k G -injection f ∗ : M ∗ → map( G, k n ) such that im( f ∗ ) is a G -inv aria n t k -subspace. Definition 1.1 0 (Elek’s dimension function) . Let M b e a finitely generated left k G - mo dule. Its dimension in t he sense of Elek is defined by choosing a left k G -surjectio n f : k G n ։ M and se tting (1.11) dim Elek kG ( M ) = dim A G  im( f ∗ )  . Theorem 1.12 (Main pr o p erties of Elek’s dimension function) . L et G b e a finitely gener ate d amenable gr oup. The definition (1.1 1) of dim Elek kG ( M ) is indep endent of the choic e of t he surje ction f , and dim Elek kG has the fol lowing pr op erties: (i) dim Elek kG ( k G ) = 1 ; (ii) F or any short exact se qu en c e of finitely gener ate d k G -mo dules 0 → M 0 → M 1 → M 2 → 0 we get dim Elek kG ( M 1 ) = dim Elek kG ( M 0 ) + dim Elek kG ( M 2 ); (iii) If the finitely gener at e d k G -mo dule M satisfies dim Elek kG ( M ) = 0 , t hen every quotient mo dule Q of M satisfies dim Elek kG ( Q ) = 0 . Pr o of. The first tw o asser tions a re prov ed in [5, Theorem 1]. Notice that the third condition do es not necessarily follow from additivity since the kernel o f the epimor phism M → Q may not be finitely generated. But the third statement is a direct consequence of the definition of Ele k’s dimensio n.  Remark 1. 13 (The dual of finitely generated k G -mo dules) . Identify the left kG -mo dule k G n with the finitely supp orted functions in map( G, k n ). Here w e vie w map( G, k n ) as a le ft k -vector space by ( af )( g ) = af ( g ), a nd the left G -action is given by ( hf )( g ) = f ( h − 1 g ) for h, g ∈ G and a ∈ k . Le t h , i : k G n × map( G, k n ) be the canonical pairing (ev aluation) of kG n and its dual map( G, k n ). If w e view an element f ∈ k G n as a finitely supp orted function G → k n (in map( G, k n )), then the pairing of 6 PETER LINNELL, W OLFGANG L ¨ UCK, AND ROMAN SA UER f ∈ k G n with l ∈ map( G, k n ) is given by h f , l i = X g ∈ G ( f ( g ) , l ( g )) , where ( , ) denotes the standar d inner pro duct in k n . F or a subs et W ⊂ k G n let W ⊥ =  f ∈ map( G, k n ) | h x, f i = 0 ∀ x ∈ W  . If M is a finitely genera ted kG - mo dule and f : k G n ։ M is a left k G -sur jection, then f ∗ : M ∗ ֒ → map( G, k n ) is a r ight k G -injection a nd im( f ∗ ) = k er( f ) ⊥ ⊆ map( G, k n ) . 2. A ppr oxima tion for finitel y presented k G -modules for El ek’s dimension function The main r e sult of this section is: Theorem 2.1. L et G b e a finitely gener ate d amenab le gr oup. Consider a se quenc e of normal sub gr oups of fi n ite index G = G 0 ⊇ G 1 ⊇ G 2 ⊇ · · · such t hat T n ≥ 0 G n = { 1 } . Then every finitely pr esente d k G -mo dule M satisfies dim Elek kG ( M ) = lim n →∞ dim k  k ⊗ kG n M  [ G : G n ] . Its pro of needs so me pre paration. Throughout, let G b e a finitely generated amena ble group. F or a ny subset S ⊂ G let k [ S ] be the k - subspace of k G ge ner ated b y S ⊂ k G . Let j [ S ] : k [ S ] → k [ G ] b e the inclusion and pr[ S ] : k G → k [ S ] b e the pr o jection g iven by pr[ S ]( g ) = ( g if g ∈ S ; 0 if g ∈ G \ S . Theorem 2.2. L et G b e a finitely gener ate d amenable gr oup. L et M b e a finitely pr esente d left k G -mo dule M with a pr esentation k G r f − → k G s p − → M → 0 . F or every subset S ⊂ G we define M [ S ] = cok er  pr[ S ] ◦ f ◦ j [ S ] : k [ S ] r → k [ S ] s  . L et ( F n ) n ≥ 0 b e a Følner se quenc e of G . Then dim Elek kG ( M ) = lim n →∞ dim k ( M [ F n ]) | F n | . Pr o of. The map f is given by r ight multiplication with a matrix A ∈ M r,s ( k G ). Viewing A as a map G → k r × s it is clea r what we mean by the suppo rt supp( A ) of A . Let R > 0 b e the diameter o f supp( A ) ∪ supp( A ) − 1 . Since lim n →∞   ∂ R F n     F n   = 0 , it is enoug h to show that for every n ≥ 1 (2.3)   dim k ( M [ F n ]) − dim k  im( p ∗ ) | F n    ≤ s ·   ∂ R F n   . THE LIMIT OF BETTI NUMBERS OF A TO WER OF FINITE COVERS 7 F or the definition of inner pro ducts ( , ) and h , i we refer to Remark 1.1 3. Define the following k -linear subspaces of map( F n , k s ): W n =  φ : F n → k s | h pr n ◦ f ◦ j n ( x ) , φ i = 0 ∀ x ∈ k [ F n ] r  ; V n =  φ : F n → k s | ∃ ¯ φ : G → k s satisfying ¯ φ | F n = φ, h f ( y ) , ¯ φ i = 0 ∀ y ∈ k G r  ; Z n =  φ : F n → k s | φ | ∂ R F n = 0  . Since dim k ( M [ F n ]) = dim k ( W n ) and dim k  im( p ∗ ) | F n  = dim k ( V n ), the desired esti- mate (2.3 ) is equiv alent to (2.4)   dim k ( W n ) − dim k ( V n )   ≤ s ·   ∂ R F n   . By additivity of dim k we o btain that dim k ( W n ∩ Z n ) ≥ dim k ( W n ) − dim k (map( F n , k s )) + dim k ( Z n ) ≥ dim k ( W n ) − s · | F n | + s · ( | F n | − | ∂ R F n | ) = dim k ( W n ) − s · | ∂ R F n | . Similarly , w e get dim k ( V n ∩ Z n ) ≥ dim k ( V n ) − s · | ∂ R F n | . T o prove (2.4) it hence suffices to show tha t W n ∩ Z n ⊂ V n ; (2.5) V n ∩ Z n ⊂ W n . (2.6) Let φ ∈ W n ∩ Z n . Extend φ b y zer o to a function ¯ φ : G → k s . Let y ∈ kG r . Then we can decomp ose y as y = y 0 + y 1 with supp( y 0 ) ⊂ F n and supp( y 1 ) ⊂ G \ F n . By definition of the r adius R it is clea r that supp( f ( y 1 )) ⊂ G \ F n ∪ ∂ R F n . Because o f φ ∈ Z n we hav e h f ( y 1 ) , ¯ φ i = 0. The fact that φ ∈ W n implies that h f ( y 0 ) , ¯ φ i = h pr n ◦ f ◦ j n ( y 0 ) , φ i = 0 . So we obtain that h f ( y ) , ¯ φ i = 0, mea ning tha t φ ∈ V n . The pr o of o f (2.6) is similar.  The following theo r em is due to W eiss. Its pro of ca n b e found in [2, P rop ositio n 5.5]. Theorem 2.7 (W eiss) . L et G b e a c ount able amenable gr oup. L et G n ⊂ G , n ≥ 1 , b e a se quenc e of normal sub gr oups of finite index with T n ≥ 1 G n = { 1 } . Then ther e exists, for every R ≥ 1 and every ǫ > 0 , an inte ger M = M ( R, ǫ ) ≥ 1 such that for n ≥ M ther e is a fundamental domain Q n ⊂ G of the c oset sp ac e G/G n such t hat   ∂ R Q n     Q n   < ǫ. Now we are r e a dy to prove Theor em 2.1 Pr o of of The or em 2.1. According to Theo rem 2.7 let ( Q n ) n ≥ 0 be a Følner sequence of G such tha t Q n ⊂ G is a fundamental domain for G/G n . Cho ose a finite pr esentation of M : k G r f − → k G s → M → 0 . Let f n = k [ G/G n ] ⊗ kG f . By right-exactness of tensor pro ducts we hav e the exact sequence k [ G/G n ] r f n − → k [ G/G n ] s → k [ G/G n ] ⊗ kG M → 0 . 8 PETER LINNELL, W OLFGANG L ¨ UCK, AND ROMAN SA UER The natura l map Q n ⊂ G → G/G n induces an isomorphism j n : k [ Q n ] → k [ G/G n ] of k - vector spaces. The map f is given b y r ight m ultiplication f = R A with a matrix A ∈ M r,s ( k G ). Viewing A as a ma p G → k r × s let s upp( A ) b e the s upp or t of A . Let R > 0 b e the diameter of supp( A ) ∪ supp( A ) − 1 (with resp ect to the fixed word metric o n G ). Then f restricts to a map f | Q n \ ∂ R Q n : k [ Q n \ ∂ R Q n ] r → k [ Q n ] s . Hence there is precis e ly one k -linea r map g for which the following diag ram of k - vector spaces co mm utes: (2.8) k [ G/G n ] r f n / / k [ G/G n ] s / / k [ G/G n ] ⊗ kG M / / 0 k [ Q n \ ∂ R Q n ] r j n | Q n \ ∂ R Q n O O f | Q n \ ∂ R Q n / / k [ Q n ] s j n ∼ = O O pr / / coker ( f | Q n \ ∂ R Q n ) g O O O O / / 0 One ea sily verifies that g is s ur jective a nd that ker( g ) ⊂ im  pr ◦ j − 1 n ◦ f n : k [ G/ G n ] r → coker( f | Q n \ ∂ R Q n )  . The map pr ◦ j − 1 n ◦ f n descends to a map pr ◦ j − 1 n ◦ f n : coker( j n | Q n \ ∂ R Q n ) → coker ( f | Q n \ ∂ R Q n ) . Note that dim k  coker ( j n | Q n \ ∂ R Q n )  = r · | ∂ R Q n | . Thu s, dim k (coker( f | Q n \ ∂ R Q n )) − dim k ( k [ G/G n ] ⊗ kG M ) = dim k ker( g ) ≤ r ·   ∂ R Q n   . By r eplacing the upper row in diag ram (2.8) by k [ Q n ] r pr[ Q n ] ◦ f ◦ j [ Q n ] − − − − − − − − − − → k [ Q n ] s → M [ Q n ] → 0 and es s entially running the s a me a rgument as befor e we obtain that dim k (coker( f | Q n \ ∂ R Q n )) − dim k (coker ( M [ Q n ])) ≤ r ·   ∂ R Q n   . Since | ∂ R Q n | [ G/G n ] = | ∂ R Q n | | Q n | n →∞ − − − − → 0 we g et that lim n →∞ dim k ( k [ G/G n ] ⊗ kG M ) [ G : G n ] exists if a nd o nly if lim n →∞ dim k ( M [ Q n ]) | Q n | exists, and in this case they are equal. Now the asser tion follows from Theorem 2.2.  THE LIMIT OF BETTI NUMBERS OF A TO WER OF FINITE COVERS 9 3. Comp aring dimensions The main r e sult of this sectio n is: Theorem 3.1 (Compa ring dimensions) . L et G b e a gr oup, let k b e a skew fi eld, and let k ∗ G b e a cr osse d pr o duct which is prime left Goldie. L et dim b e any dimension function which assigns to a finitely gener ate d left k ∗ G -mo dule a nonne gative r e al nu mb er and satisfies (i) dim( k ∗ G ) = 1 . (ii) F or every short exact se quenc e 0 → M 0 → M 1 → M 2 → 0 of finitely gener ate d left k ∗ G -mo dules, we get dim( M 1 ) = dim( M 0 ) + dim( M 2 ) . (iii) If the fi nitely gener ate d left k ∗ G -mo dule M satisfies dim( M ) = 0 , then every quotient mo dule Q of M satisfies dim( Q ) = 0 . Then for every finitely pr esente d left k ∗ G -mo dule M , we get dim( M ) = dim Ore k ∗ G ( M ) . Pr o of. Let S denote the non-zer o-divisor s of k ∗ G . W e hav e to show that for all r , s ∈ N and every r × s matrix A with entries in k ∗ G (3.2) dim Ore k ∗ G  coker  r A : S − 1 k ∗ G r → S − 1 k ∗ G s  = dim  coker  r A : k ∗ G r → k ∗ G s  , where r A denotes the mo dule homomo rphism given by right multiplication with A . First note that we may assume that r = s . Indeed if r < s , replace A with the s × s matrix which is A for the first r rows, and has 0’s on the bo ttom s − r rows. On the other hand if r > s , replace A with the r × r matrix B with entries ( b ij ) which is A for the first s columns, and has b ij = δ ij if i > s , where δ ij is the Kr oneck er delta. W e will often use the ob vious lo ng exact s equence a sso ciated to homomorphisms f : M 0 → M 1 and g : M 1 → M 2 (3.3) 0 → k er( f ) → ker( g ◦ f ) → ker( g ) → coker( f ) → coker ( g ◦ f ) → coker( g ) → 0 . W e now assume that A is an r × r matr ix. Note that eq uation (3.2) is true if A is invertible ov er S − 1 k ∗ G ; this is b ecause then ker r A = 0 (whether A is considered a s a matrix ov er k ∗ G o r S − 1 k ∗ G ). Next observe that if U ∈ M r ( k ∗ G ) which is inv ertible ov er M r ( S − 1 k ∗ G ), then equatio n (3.2) ho lds for A if and o nly if it ho lds for AU , and also if and only if it hold for U A . This follows from (3.3), ker U = 0, dim(co ker U ) = dim Ore k ∗ G (coker U ) = 0 , and in the sec o nd case we us e the third prop er t y of dim. W e may write S − 1 k ∗ G = M d ( D ) for some d ∈ N and some skew field D . By applying the Morita equiv a lence from M d ( D ) to D and back and do ing Gaussian e limina tion ov er D w e see that there are inv ertible matric e s U, V ∈ M r d ( S − 1 k ∗ G ) such that U dia g( A, . . . , A ) V = J , where ther e are d A ’s and J is a matr ix of the form diag(1 , . . . , 1 , 0 , . . . , 0). Now choose u, v ∈ S such that uU , v V ∈ M r d ( k ∗ G ). Then ( uU ) diag( A, . . . , A )( V v ) = uJ v , and the result fo llows.  Theorem 3.4 (Compa r ing Elek ’s dimensio n a nd the Or e dimens ion) . L et G b e a finitely gener ate d gr oup and let k b e a skew field. Su pp ose that k G is a prime left Goldie ring. Then for any finitely pr esente d left k G -mo dule M dim Elek kG ( M ) = dim Ore kG ( M ) . Pr o of. This follows from Theorem 3.1 and Theo r em 1 .12.  10 PETER LINNELL, W OLFGANG L ¨ UCK, AND ROMAN SA UER 4. P roof of the main theorem Pr o of of The or em 0.2. (i) In the first step we reduce the claim to the case, where G is finitely generated. Consider a finitely pre sented left k G -mo dule M . Cho o se a matrix A ∈ M r,s ( k G ) such that M is k G -isomorphic to the cokernel of r A : k G r → k G s . Since A is a finite matrix and each elemen t in k G has finite suppo rt, w e can find a finitely g e nerated subgro up H ⊆ G such that A ∈ M r,s ( k H ). Both k G a nd k H are pr ime left Go ldie by Lemma 1.4 and Theorem 1.6. Co nsider the finitely presen ted kH -mo dule N := cok er  r A : k H r → kH s  . Then M = k G ⊗ kH N . W e can a lso co nsider the Ore lo caliza tion T − 1 k H for T the set o f non-zero- divisors of k H . Put H n = H ∩ G n . W e obtain a res idual chain ( H n ) n ≥ 0 of H and hav e: dim Ore kG ( M ) = dim S − 1 kG  S − 1 k G ⊗ kG M  = dim S − 1 kG  S − 1 k G ⊗ kG k G ⊗ kH N  = dim S − 1 kG  S − 1 k G ⊗ T − 1 kH T − 1 k H ⊗ kH N  = dim T − 1 kH  T − 1 k H ⊗ kH N ) = dim Ore kH ( N ) . W e co mpute dim k  k ⊗ kG n M  [ G : G n ] = dim k  k [ G/G n ] ⊗ kG M  [ G : G n ] = dim k  k [ G/G n ] ⊗ kG k G ⊗ kH N  [ G : G n ] = dim k  k [ G/G n ] ⊗ k [ H/H n ] k [ H /H n ] ⊗ kH N  [ G : G n ] = [ G/G n : H/ H n ] · dim k  k [ H /H n ] ⊗ kH N  [ G : G n ] = [ G/G n : H/ H n ] · dim k  k ⊗ kH n N  [ G/G n : H/ H n ] · [ H : H n ] = dim k  k ⊗ kH n N  [ H : H n ] . Therefore the claim holds for M over k G if it holds for N ov er k H . Hence we ca n a ssume without loss of gener ality that G is finitely gener ated. Now a pply Theo rem 2 .1 and Theor em 3 .4. (ii) W e obtain fr om a dditivit y , the exactness of the functor S − 1 k G ⊗ kG and the right exactness o f the functor k ⊗ kG that dim Ore kG  H i ( C ∗ )  = dim Ore kG  coker ( c i +1 )  + dim Ore kG  coker ( c i )  − dim Ore kG  C i − 1  , dim k  H i ( k ⊗ kG n C ∗ )  = dim k  k ⊗ kG n coker ( c i +1 )  + dim k  k ⊗ kG n coker ( c i )  − dim k  k ⊗ kG n C i − 1  . Hence the claim follo ws from asser tion (i) a pplied to the finitely presented k G -mo dules coker ( c i +1 ), coker ( c i ) and C i − 1 . (iii) This fo llows fro m assertion (ii) applied to the cellular chain complex of X .  THE LIMIT OF BETTI NUMBERS OF A TO WER OF FINITE COVERS 11 5. E xtension to the vir tuall y torsionfree case Next w e explain how Theorem 0.2 c an b e extended to the vir tually to r sionfree c a se. F or the re ma inder of this sectio n let k b e a skew field, let G b e an amenable gr o up which po ssesses a subgroup H o f finite index with ∆ + ( H ) = 1, and let k ∗ G be a cr ossed pro duct such that k ∗ H is a left Goldie ring. W e define the virtual Or e dimensio n of a k ∗ G -mo dule M by (5.1) vdim Ore k ∗ G ( M ) = dim Ore k ∗ H  res k ∗ H k ∗ G M  [ G : H ] , where res k ∗ H k ∗ G M is the k ∗ H -mo dule obtained fro m the k ∗ G -mo dule M by restr icting the G -action to H . W e hav e to s how that this is independent o f the choice o f H . Since every subgr oup o f finite index contains a no r mal subg roup of finite index, it is enoug h to show that if K is a normal subgr oup of finite index in H and K ≤ H ≤ G w ith H torsion fre e , then for every k ∗ H -module N , (5.2) dim Ore k ∗ K  res k ∗ K k ∗ H N  [ H : K ] = dim Ore k ∗ H ( N ) . Let T deno te the set of no n-zero-div is ors of k ∗ H and write S = ( k ∗ K ) ∩ T . Note that ∆ + ( K ) = 1 , so k ∗ K is still a prime left Go ldie ring and hence the ring S − 1 k ∗ K exists. Then S − 1 k ∗ H ∼ = ( S − 1 k ∗ K ) ∗ [ H/K ] and there is a na tural ring monomorphis m θ : S − 1 k ∗ H ֒ → T − 1 k ∗ H . Since S − 1 k ∗ K is a matrix ring over a skew field by Theorem 1.5, we see that ( S − 1 k ∗ K )[ H/K ] is a n Artinian r ing, b eca use H/ K is finite. But every element of T is a non-zer o-divisor in ( S − 1 k ∗ K )[ H/K ], and since every non-zero- divisor in an Artinian ring is in vertible (compare [16, Exercise 22 of Chapter 15 on p. 16]), we see that every element of T is inv ertible in S − 1 k ∗ H and we conclude tha t θ is on to and hence is an isomorphism. W e deduce that dim S − 1 k ∗ K ( T − 1 k ∗ H ) = [ H : K ] and that the natural map S − 1 N → T − 1 N induced b y s − 1 n 7→ s − 1 n is a n isomorphism. This prov es (5.2). Theorem 5.3 (Extension to the virtually tor sionfree case) . L et G b e an amenabl e gr oup which p ossesses a sub gr oup E ⊆ G of finite index such t hat k E is left Goldie and ∆ + ( E ) = 1 , and let k b e a skew field. Then assertions (i), (ii) and ( iii) of The or em 0.2 r emain true, pr ovid e d we r eplac e dim Ore kG by vdim Ore kG everywher e. Pr o of. It s uffices to prov e the claim for asse rtion (i) since the pro of in Theorem 0.2 that it implies the other tw o a ssertions applies also in this more genera l situatio n. Let ( G n ) n ≥ 0 be a r esidual chain o f G . T o pr ov e the result in gener al, we may assume that G is finitely generated. Since k E is le ft Goldie and [ G : E ] < ∞ , the group ring k G is a ls o left Goldie. F urther, every k G n is left Goldie. Since ∆ + ( G ) is finite (its order is b ounded by [ G : E ]), there exists N ∈ N such that G N ∩ ∆ + ( G ) = 1, and then ∆ + ( G N ) = 1 and G i ⊆ G N for all i ≥ N . Set H = G N , s o that k H is prime by Lemma 1.4. Then for a finitely presented k H -mo dule L , dim Ore kH ( L ) = lim n →∞ dim k  k ⊗ k [ G n ∩ H ] L  [ H : H ∩ G n ] 12 PETER LINNELL, W OLFGANG L ¨ UCK, AND ROMAN SA UER by Theor ems 2.1 and 3.1. W e hav e [ G : G n ] = [ G : H ] · [ H : H ∩ G n ] for n ≥ N . This implies for e very finitely pr esented k G -mo dule M vdim Ore kG ( M ) = dim Ore kH  res kH kG M  [ G : H ] = lim n →∞ dim k  k ⊗ k [ H ∩ G n ] res kH kG M  [ G : H ] · [ H : H ∩ G n ] = lim n →∞ dim k  k ⊗ kG n M  [ G : G n ] .  Remark 5.4. B ecause of Theo rem 1.6, The o rem 0.2 is true in the case k is a skew field and G is a n e le mentary a menable group in whic h the order s of the finite subgroups are b ounded (clearly ∆ + ( G n ) = 1 for sufficiently large n ). In particular Theorem 0.2 is true fo r any virtually torsionfree elementary a menable g roup. 6. E xamples Remark 6.1 . Let ( G n ) n ≥ 0 be a residua l chain of a group G . Let X b e a finite free G - C W - complex. Let k be a field o f c har acteristic char( k ). F or a prime p denote b y F p the field o f p elements. Then we conclude fro m the univ ers al co efficient theorem dim k  H i ( G n \ X ; k )  = dim Q  H i ( G n \ X ; Q )  char( k ) = 0; dim k  H i ( G n \ X ; k )  = dim F p  H i ( G n \ X ; F p )  p = c har( k ) 6 = 0; dim F p  H i ( G n \ X ; F p )  ≥ dim Q  H i ( G n \ X ; Q )  . In particular we co nclude fro m Rema r k 0.3 that lim inf n →∞ dim k  H i ( G n \ X ; k )  [ G : G n ] ≥ lim n →∞ dim k  H i ( G n \ X ; Q )  [ G : G n ] = b (2) i ( X ; N ( G )) , where the latter term denotes the i -th L 2 -Betti num b er of X . In particular we ge t from Theorem 0.2 for a tor s ionfree amena ble gr oup G with no zero- divisors in k G that dim Ore kG  H i ( X ; k )  = lim n →∞ dim k ( H i ( G n \ X ; k )) [ G : G n ] ≥ lim n →∞ dim k ( H i ( G n \ X ; Q )) [ G : G n ] = b (2) i ( X ; N ( G )) = dim Ore C G  H i ( X ; C )  . This inequa lity is in g eneral no t an equa lit y as the next example s hows. Example 6.2. Fix a n in teger d ≥ 2 and a prime num b er p . Let f p : S d → S d be a map of degr ee p and denote b y i : S d → S 1 ∨ S d the ob vious inclusion. Let X b e the finite C W -complex obta ine d from S 1 ∨ S d by attaching a ( d + 1)-cell with attaching ma p i ◦ f d : S d → S 1 ∨ S d . Then π 1 ( X ) = Z . Let e X b e the universal c overing of X which is a finite free Z - C W -complex. Denote by X n the cov ering of X a sso ciated to n · Z ⊆ Z . The cellular Z C -chain complex of e X is concentrated in dimension ( d + 1), d a nd 1 a nd 0, the ( d + 1)-th differe ntial is multiplication with p and the first differ e ntial is multiplication with ( z − 1) fo r a g enerator z ∈ Z 0 → · · · → Z [ Z ] p − → Z [ Z ] → · · · → Z [ Z ] z − 1 − − → Z [ Z ] . If the characteris tic o f k is different fro m p , one easily chec ks that H i ( C ∗ ) = 0 and dim Ore k Z  H i ( e X ; k )  = 0 for i ∈ { d, d + 1 } . THE LIMIT OF BETTI NUMBERS OF A TO WER OF FINITE COVERS 13 If p is the characteristic of k , then H i ( C ∗ ) = k Z and dim Ore k Z  H i ( e X ; k )  = 1 for i ∈ { d, d + 1 } . Hence dim Ore kG  H i ( e X ; k )  do es dep end on k in general. Example 6 .3. Let G be a torsionfree amena ble group s uch that k G has no zero-div isors. Let S 1 → X → B b e a fibra tion of connec ted C W -complexes such that X ha s fundament al group π 1 ( X ) ∼ = G and π 1 ( S 1 ) → π 1 ( X ) is injective. Then (6.4) dim Ore kG  H i ( e X ; k )  = 0 for e very i ≥ 0. Let S = k G − { 0 } a nd S 0 = k Z − { 0 } . B y lo o king a t the cellular chain complex one directly sees that H i  e S 1 , S − 1 0 k Z  = 0 ∀ i ≥ 0 , th us H i  e S 1 , S − 1 k G  = S − 1 k G ⊗ S − 1 0 k Z H i  e S 1 , S − 1 0 k Z  = 0 for every i ≥ 0 . 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