Induced quasi-actions: a remark

INDUCED QUASI-A CTIONS: A REMARK BRU CE KLEINER AND BERNHARD LEEB 1. Introduction In this note w e observ e that t he not io n o f an induced represen tation has a n analog for quasi-actio ns, and giv e some applications. W e will use the definitions and notation fro m [KL01]. 1.1. Induced qua si-actions an d their prop erties. Let G b e a gro up and { X i } i ∈ I b e a finite collection of un b ounded metric spaces. Definition 1.1. A quasi-action G ρ y Q i X i pr eserves the pr o duct structur e if each g ∈ G acts b y a product of quasi-isometries , up to uniformly b ounded error. Note that we allow the quasi-isometries ρ ( g ) to p erm ute t he factors, i.e. ρ ( g ) is uniformly close to a map of the form ( x i ) 7→  φ σ − 1 ( i ) ( x σ − 1 ( i ) )  with a p ermutation σ of I and quasi-isometries φ i : X i 7→ X σ ( i ) . Asso ciated to ev ery quasi-action G ρ y Q i X i preserving pro duct structure is the action G ρ I y I corres p onding to the induced p erm uta- tion of the factors; this is well-define d because the X i ’s are unbounded metric spaces . F or each i ∈ I , the stabilizer G i of i with respect t o ρ I has a quasi-action G i y X i b y restriction of ρ . It is w ell-defined up to equiv a lence in the sense o f [KL01, Definition 2.3]. If the permutation action ρ I is tr ansitive , all factors X i are quasi- isometric to eac h other, and the restricted quasi-actions G i y X i are quasi-conjugate (when identifying differen t stabilizers G i b y inner au- tomorphisms o f G ). The main result of t his note is that in this case any of the quasi-actions G i y X i determines ρ up to quasi-conjugacy , and moreo ver any quasi-conjugacy class may arise as a restricted action. Theorem 1.2. L et G b e a gr oup, H b e a finite index sub gr oup, and H α y X b e a q uasi - action of H on an unb ounde d m etric sp ac e X . Date : July 30, 200 7. The first a uthor was pa rtially supp orted by NSF Grant DMS 07015 15. 1 2 BRUCE KLEINER AND BERNHAR D LEEB Then ther e exists a quasi-action G β y Q i ∈ G/H X i pr eserving pr o duct structur e, wher e (1) Each f a ctor X i is quasi-isom etric to X . (2) The asso ciate d action G β G/H y G/H is the natur al action by left multiplic ation. (3) The r estriction of β to a quasi-action of H on X H is quasi- c onjugate to H α y X . F urthermor e, ther e is a unique such quasi - a ction β pr eserving the pr o d - uct struct ur e, up to quas i - c o njugacy b y a pr o duct quasi-i s ometry. Fi- nal ly, if α is an isometric action, then the X i may b e taken isometric to X an d β may b e taken to b e an isome tric action . Definition 1.3. Let G , H and H y X b e as in Theorem 1.2. The quasi-action β is called the quasi - a ction induc e d by H y X . As a b ypro duct of the main construction, we get the following: Corollary 1.4. If G ρ y X is an ( L, A ) -quasi-action on an arbitr ary metric sp ac e X , then ρ is ( L, 3 A ) -quasi-c onjugate to a c anonic al ly de- fine d isometric action G y X ′ . 1.2. Applications. The implication of Theorem 1.2 is tha t in order to quasi-conjugate a quasi-action on a pro duct to an isometric action, it suffices to quasi-conjugate the factor quasi-a ctio ns to isometric actions. W e b egin with a special case: Theorem 1.5. L et G ρ y X b e a c o b ounde d quasi-action on X = Q i X i , wher e e ac h X i is either an irr e ducible symm e tric s p a c e of non- c omp act typ e, or a thick irr e ducible Euclide an buildi n g of r an k at le ast two, with c o c omp act Weyl gr oup. Then ρ is quasi - c onjugate to an iso- metric a c tion on X , after suitable r esc aling of the metrics on the factors X i . R emarks • Theorem 1.5 was stated incorrectly as Corollary 4.5 in [K L01]. The pro of g iv en there w as w as only v alid for quasi-actions whic h do not p erm ute the f a ctors. • Rescaling of the factors is necessary , in general: if one t a k es the pro duct of tw o copies of H 2 where t he facto r s are scaled to ha v e differen t curv ature, then a quasi-action whic h p erm utes the f actors will not be quasi-conjugate to a n isometric action. INDUCED QUASI-ACTIONS: A REMARK 3 W e no w consider a more general situation. Let G α y Q i ∈ I X i b e a quasi-action, where eac h X i is one of the followin g four t yp es of spaces: (1) An irreducible symmetric space of noncompact t yp e. (2) A thick irreducible Euclidean building of rank/dimension ≥ 2, with co compact W eyl group. (3) A b ounded v alence bush y tree in the sense of [MSW03]. W e recall that a tree is b ushy if eac h of its p oints lies within uni- formly b ounded distance f r o m a v ertex ha ving at least three un b ounded complemen tary comp onen ts. (4) A quasi-isometrically rigid Gro mov hy p erb olic space whic h is of coarse type I in the sense of [KK L 98, sec. 3] (see the remarks b e- lo w). A space is quasi-isometric al ly rigid if eve ry ( L, A )-quasi- isometry is at distance at most D = D ( L, A ) fr om a unique isometry . By [KK L 9 8, Theorem B], the quasi-action preserv es pro duct structure, and hence w e ha v e an induced p ermutation action G y I . Let J ⊂ I b e the set of indices i ∈ I suc h that X i is either a real hyperb olic space H k for some k ≥ 4, a complex h yp erb o lic space CH l for some l ≥ 2, o r a b ounded v alence bush y tree. Generalizing Theorem 1.5 we obtain: Theorem 1.6. If the quasi-action G j y X j is c ob ounde d for e ach j ∈ J , then α is quasi-c onjugate by a pr o duct quasi-isometry to a n iso metric action G α ′ y Q i ∈ I X ′ i , wher e for every i , X ′ i is quasi-isometric to X i , and pr e cisely one of the fol lowing holds: (1) If X i is not a b ounde d valenc e bushy tr e e, then X ′ i is isometric to X i ′ for some i ′ in the G -orbit G ( i ) of i . (2) If X i is a b ounde d valenc e bushy tr e e, then so is X ′ i . As in the previous corollary , it is necessary to permit X ′ i to b e noni- sometric to X i . Moreo ver, there ma y b e factors X i and X j of type (4) lying in the same G -orbit, but whic h a re no t ev en homothetic, so it is not sufficien t to allow rescaling of f actors. Pr o of. W e first assume that the a ction G y I is transitiv e. Pic k n ∈ I . Then t he quasi-action G n y X n is quasi-conjugate to an isometric action G n y X ′ n , where X ′ n is isometric to X n unless X n is a b ounded v alence bush y tree, in whic h case X ′ n is a b ounded v alence bush y tree but not necessarily isometric to X n ; this follows from: • [Hin90, Gab92, CJ94, Mar06] when X n is H 2 . Note that an y quasi- action on H 2 is quasi-conjuga te to an isometric actio n. 4 BRUCE KLEINER AND BERNHAR D LEEB • [Sul81, Gro, T uk86, P an8 9 , Cho96] when X n is a r a nk 1 symmetric space other than H 2 . Note tha t Sulliv an’s theorem implies that any quasi-action on H 3 is quasi-conjugate to an isometric action. Also, the pro of giv en in Cho w’s pa p er on the complex h yp erb olic case cov ers arbitrary cob ounded quasi-actions, eve n though it is o nly stated for discrete cob ounded quasi-a ctions. • [K L97, Lee00] when X n is an irreducible symmetric space or Eu- clidean building of rank a t least 2. • [MSW03] when X n is a b ounded v alence bush y tree. By Theorem 1.2, the asso ciated induced quasi-action of G is quasi- conjugate t o the original quasi-action G y Q i ∈ I X i b y a pro duct quasi-isometry , and we are done. In t he general case, for each orbit G ( i ) ⊂ I of the action G y I , w e ha ve a w ell-defined asso ciated quasi-action G y Q j ∈ G ( i ) X j for whic h the theorem has already b een established, and w e obtain the desired isometric action G y Q i ∈ I X ′ i b y taking pro ducts.  Corollary 1.7. L et { X i } i ∈ I b e as ab ove, and supp ose G is a fin itely gener ate d gr oup quasi-isometric to the pr o duct Q i ∈ I X i . Then G ad- mits a discr ete, c o c om p act, isometric action on a pr o duct Q i ∈ I X ′ i , wher e fo r every i , X ′ i is quasi-isometric to X i , and pr e ci s e ly one of the fol lowing holds: (1) X i is not a b o und e d valenc e bushy tr e e, and X ′ i is isometric to X i ′ for some i ′ in the G -orbit G ( i ) ⊂ I of i . (2) Both X i and X ′ i ar e b ounde d valen c e bushy tr e es. Pr o of. Suc h a group G admits a discrete, cob o unded quasi-action on Q i ∈ I X i . Theorem 1 .6 furnishes the desired isometric action G y Q i X ′ i .  R emarks. • Corollary 1.7 refines earlier results [Ahl02, KL01, MSW03]. • A prop er Gromov h yp erb olic space with co compact isometry group is of coarse type I unless it is quasi-isometric to R [KKL98, Sec. 3]. • The classification of the four differen t types of spaces ab o v e is quasi-isometry in v ariant, with one exception: a space of type (1) will also b e a space of t yp e (4) iff it is a quasi-isometrically INDUCED QUASI-ACTIONS: A REMARK 5 rigid rank 1 symmetric space (i.e. a quaternionic h yp erb olic space or the Cayley h yp erb olic plane [P an89]). See L emma 3.1. • Tw o irreducible symmetric spaces ar e quasi-isometric iff t hey are isometric, up to rescaling [Mos73, P an89, KL97]. Tw o Eu- clidean buildings as in (2) ab ov e are quasi-isometric iff they ar e isometric up to rescaling [KL97, Lee00]. 2. The const r uction of induced quasi-a ctions The construction of induced quasi-actions is a direct imitation of one of the standard constructions of induce d represen tations. W e no w review this for the con v enience o f the reader. Let H b e a subgroup of some g roup G , a nd supp ose α : H y V is a linear represen tation. Then w e ha ve an action H y G × V where ( h, ( g , v )) = ( g h − 1 , hv ). Let E := ( G × V ) /H b e the quotient. There is a natural pro jection map π : E → G/H whose fib ers are copies o f V ; this w ould b e a v ector bundle o ver the discrete space G/H if V w ere endo we d with a to p ology . The action G y G × V b y left translation on t he first factor descends to E , and comm utes with the pro jection map π . Moreov er, it preserv es the linear structure on the fib ers. Hence there is a represen ta tion of G on the v ector space of sections Γ( E ), and this is the repres en ta tion of G induced b y α . W e use the terminology of [KL01, Sec. 2]. (Ho w ev er, we replace quasi-isometric al ly c onjugate b y the shorter and more accurate term quasi-c onjugate .) W e will w ork with generalize d metrics taking v alues in [0 , + ∞ ]. A finite c omp on ent of a generalized metric space is an equiv alence class o f p oin ts with pairwise finite distances. Clearly , quasi-isometries resp ect finite comp onen ts. Let { X i } i ∈ I b e a finite collection of un b ounded metric spaces in the usual sense, i.e. the metric on each X i tak es only finite v alues. On their pro duct Q i ∈ I X i w e consider the natural ( L 2 -)pro duct metric. On their disjoin t union ⊔ i ∈ I X i w e consider the g eneralized metric whic h induces the orig inal metric o n each comp onent X i and g ives distance + ∞ to an y pair of p oints in differen t comp onen t s. W e observ e that a quasi-isometry Q i ∈ I X i → Q i ∈ I X ′ i preserving the pro duct structure g iv es rise to a quasi-isometry ⊔ i ∈ I X i → ⊔ i ∈ I X ′ i , w ell-defined up t o b ounded error, and vice v ersa. Th us equiv a lence 6 BRUCE KLEINER AND BERNHAR D LEEB classes of quasi-actio ns α : G y Q i ∈ I X i preserving t he pro duct struc- ture correspond o ne-to-one t o quasi-actions β : G y ⊔ i ∈ I X i . In what follo ws w e will prov e the disjoin t union analog of Theorem 1.2 . (The index of H can b e arbitrary from now on.) Lemma 2.1. Supp os e that Y is a gene r alize d metric sp a c e and that G y Y i s a quasi-action such that G acts tr ansitively on the set of finite c omp on e nts of Y . L et Y 0 b e one of the finite c omp onents and H its stabilizer in G . Then the r estricte d action H y Y 0 determines the action G y Y up to quasi-c onjugacy. Pr o of. If G y Y ′ is another quasi-action, Y ′ 0 is a finite comp onen t with stabilizer H , then any quasi-conjuga cy b etw een H y Y 0 and H y Y ′ 0 extends in a straigh tforw ard w ay t o a quasi-conjugacy b etw een G y Y and G y Y ′ .  W e will now show ho w to reco ver the G -quasi-a ction from the H - quasi-action b y quasifying the construction of induced actions as de- scrib ed ab o v e. Definition 2.2. An ( L, A ) -c o arse fibr ation ( Y , F ) consists of a ( gen- eralized) metric space Y and a family F of subsets F ⊂ Y , the c o a rs e fib ers , with the follo wing prop erties: (1) The unio n ∪ F ∈F F of all fib ers has Hausdorff distance ≤ A from Y . (2) F or an y t wo fib ers F 1 , F 2 ∈ F holds d H ( F 1 , F 2 ) ≤ L · d ( y 1 , F 2 ) + A ∀ y 1 ∈ F 1 . W e a lso sa y that F is a coa r se fibration o f Y . Note that the coarse fib ers are no t required to b e disjoint. It follo ws from part (2) of the definition that d H ( F 1 , F 2 ) < + ∞ if and o nly if F 1 and F 2 meet the same finite comp onen t o f Y . W e will equip the “base space” F with the Hausdorff metric. Lemma 2.3. If H y Y is a n ( L, A ) -q uasi-action then the c ol le ction of quasi-orbits O y := H · y form s a n ( L, 3 A ) -c o arse fib r ation o f Y . Pr o of. F or h, h 1 , h 2 ∈ H and y 1 , y 2 ∈ Y w e ha v e d  hy 1 , ( hh − 1 1 h 2 ) y 2  ) ≤ d  ( hh − 1 1 )( h 1 y 1 ) , ( hh − 1 1 )( h 2 y 2 )  )+2 A ≤ L · d ( h 1 y 1 , h 2 y 2 )+3 A and so d ( O y 1 , O y 2 ) ≤ L · d ( h 1 y 1 , O y 2 ) + 3 A. INDUCED QUASI-ACTIONS: A REMARK 7  Let ( Y , F ) and ( Y ′ , F ′ ) b e coa rse fibrations. W e sa y that a map φ : Y → Y ′ quasi-r esp e cts the coarse fibrations if the image of each fib er F ∈ F is uniformly Hausdorff close to a fib er F ′ ∈ F ′ , d H ( φ ( F ) , F ′ ) ≤ C . The map φ then induces a map ¯ φ : F → F ′ whic h is we ll-defined up to b o unded error ≤ 2 C . Observ e that if φ is an ( L, A )-quasi-isometry then ¯ φ is an ( L, A + 2 C )- quasi-isometry . W e sa y that a quasi-action ρ : G y Y quasi-r esp e cts a coarse fibra- tion F if all maps ρ ( g ) quasi-resp ect F with uniformly bo unded error. The quasi-a ctio n ρ then descends to a quasi-action ¯ ρ : G y F whic h is unique up to equiv alence (cf. [KL01, Definition 2.3]). W e apply these general remarks to the follo wing situation in order to obta in our main construction. Let G b e a group, H < G a subgroup (of arbitra ry index) a nd H α y X an ( L, A )-quasi-action. Let Y = G × X where G is giv en the metric d ( g 1 , g 2 ) = + ∞ unless g 1 = g 2 . That is, Y consists of | G | finite comp onen ts eac h of whic h is a copy of X . The quasi-action α give s rise to a pro duct quasi-action H ρ H y Y via ρ H  h, ( g , x )  = ( g h − 1 , hx ) . W e denote by F H the coar se fibration of Y b y H -quasi-orbits. The isometric G - a ction giv en b y ˜ ρ G  g ′ , ( g , x )  = ( g ′ g , x ) comm utes with ρ H . A s a consequence, ˜ ρ G descends to an isometric action (2.4) ˆ β := ¯ ρ G : G y F H . If H = G then α is quasi-conjugate to ˆ β via the quas i-isometry x 7→ ρ H ( H ) · ( e, x ). In general, the finite comp onen ts of F H corresp ond to the left H - cosets in G . More precisely , g H corresp onds to ∪ x ∈ X ρ H ( H ) · ( g , x ), that is, to the union of ρ H -quasi-orbits con tained in g H × X . H stabilizes the finite comp onent ∪ x ∈ X ρ H ( H ) · ( e, x ). The action of H on this comp onen t is quasi-conjugate to α . As remark ed in t he b eginning of this section, ˆ β is the unique G - quasi- action up to quasi-conjugacy suc h that G acts transitiv ely on finite comp onen ts and suc h tha t H is the stabilizer of a finite comp onent and the restricted H -quasi-action is quasi-isometrically conjugat e to α . 8 BRUCE KLEINER AND BERNHAR D LEEB P assing bac k from disjoin t unions to pro ducts w e obtain Theorem 1.2. 3. Quasi-isometrie s and the classifica tion into type s (1)-(4) W e now prov e: Lemma 3.1. Supp os e Y and Y ′ ar e sp ac es of one of typ es (1)-(4) as in The or em 1.6. If Y is quasi-isometric to Y ′ , then they ha v e the same typ e, unless on e is a quasi-isom e tric al ly rigid r ank 1 symmetric sp a c e, and the other is of typ e (4). Pr o of. First supp ose one of the spaces is not Gromov hy p erb olic. Since Gromov h yp erb olicit y is quasi-isometry inv ar ia n t, b ot h spaces m ust b e higher ra nk space of either of type ( 1 ) or (2). But b y [KL97 ], t w o irreducible symmetric spaces or Euclidean buildings of r ank at least t wo ar e quasi-isometric iff they are homothetic. Th us in this case they m ust ha v e the same type. No w a ssume b oth spaces are Gr o mo v h yp erb olic. Then ∂ Y and ∂ Y ′ are homeomorphic. If Y is a b ounded v alence bush y tree, then it is w ell-known tha t Y is quasi-isometric to a tr iv alen t tree, and ∂ Y is homeomorphic to a Can tor set. There fore Y cannot b e quasi-isometric to a space of t yp e (1), since the b oundary of a Gromov h yp erb olic symme tric space is a sphere. Also, the quasi-isometry g r o up o f a triv alen t tree T has an induced actio n on the space of triples in ∂ T w hic h is not prop er, and hence it cannot be quasi-isometric to a space of t yp e (4). If Y is a h yp erb olic or complex hyperb olic space, then the induced action of Q I( X ) on the space of triples in ∂ X is not prop er, and hence Y cannot be quasi-isometric to a space of t yp e (4). 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M ¨ unchen, Theresienstr. 39, D-80333 M ¨ unchen E-mail addr ess : b.l@lm u.de