K-theory and actions on Euclidean retracts · K-THEORYANDACTIONSONEUCLIDEANRETRACTS 1063 2...

P . I . C . M . – 2018Rio de Janeiro, Vol. 2 (1059–1080)

K -THEORY AND ACTIONS ON EUCLIDEAN RETRACTS

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Abstract

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms ofactions on Euclidean retracts and applications of these to GLn(Z), relative hyperbolicgroups and mapping class groups.

Introduction

Motivated by surgery theoryHsiang [1984]made a number of influential conjectures aboutthe K-theory of integral group rings Z[G] for torsion free groups G. These conjecturesoften have direct implications for the classification theory of manifolds of dimension � 5.A good example is the following. An h-cobordism is a compact manifold W that hastwo boundary componentsM0 andM1 such that both inclusionsMi ! W are homotopyequivalences. The Whitehead group Wh(G) is the quotient ofK1(Z[G]) by the subgroupgenerated by the canonical units ˙g, g 2 G. Associated to an h-cobordism is an in-variant, the Whitehead torsion, in Wh(G), where G is the fundamental group of W . Aconsequence of the s-cobordism theorem is that for dimW � 6, an h-cobordism W istrivial (i.e., isomorphic to a productM0�[0; 1]) iff its Whitehead torsion vanishes. Hsiangconjectured that forG torsion free Wh(G) = 0, and thus that in many cases h-cobordismsare products.

The Borel conjecture asserts that closed aspherical manifolds are topologically rigid,i.e., that any homotopy equivalence to another closed manifold is homotopic to a homeo-morphism. The last step in proofs of instances of this conjecture via surgery theory uses avanishing result forWh(G) to conclude that an h-cobordism is a product and that thereforethe two boundary components are homeomorphic.

Farrell and Jones [1986] pioneered amethod of using the geodesic flow on non-positivelycurved manifolds to study these conjectures. This created a beautiful connection between

The work has been supported by the SFB 878 in Münster.MSC2010: primary 18F25; secondary 19G24, 20F67, 20F65.Keywords: Farrell-Jones Conjecture,K- and L-theory of group rings.

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K-theory and dynamics that led Farrell and Jones [1993b], among many other results, to aproof of the Borel Conjecture for closed Riemannian manifolds of non-positive curvatureof dimension � 5. Moreover, Farrell and Jones [1993a] formulated (and proved manyinstances of) a conjecture about the structure of the algebraic K-theory (and L-theory)of group rings, even in the presence of torsion in the group. Roughly, the Farrell-JonesConjecture states that the main building blocks for the K-theory of Z[G] is the K-theoryof Z[V ] where V varies over the family of virtually cyclic subgroups of G. It implies anumber of other conjectures, among them Hsiang’s conjectures, the Borel Conjecture indimension � 5, the Novikov Conjecture on the homotopy invariant of higher signatures,Kaplansky’s conjecture about idempotents in group rings, see Lück [2010] for a summaryof these and other applications.

My goal in this note is twofold. The first goal is to explain a condition formulated interms of existence of certain actions of G on Euclidean retracts that implies the Farrell-Jones Conjecture for G. This condition was developed in A. Bartels and Lück [2012c]and A. Bartels, Lück, and Reich [2008b] where the connection between K-theory anddynamics has been extended beyond the context of Riemannian manifolds to prove theFarrell-Jones Conjecture for hyperbolic and CAT(0)-groups. The second goal is to outlinehow this condition has been used in joint work with Lück, Reich and Rüping and withBestvina to prove the Farrell-Jones Conjecture for GLn(Z) and mapping class groups. Acommon difficulty for both families of groups is that their natural proper actions (on theassociated symmetric space, respectively on Teichmüller space) is not cocompact. In bothcases the solution depends on a good understanding of the action away from cocompactsubsets and an induction on a complexity of the groups. As a preparation for mappingclass groups we also discuss relatively hyperbolic groups.

The Farrell-Jones Conjecture has a prominent relative, the Baum-Connes Conjecturefor topologicalK-theory of groupC �-algebras Baum andConnes [2000] andBaum, Connes,and Higson [1994]. The two conjectures are formally very similar, but methods of proofsare different. In particular, the conditions discussed in Section 2 are not known to implythe Baum-Conjecture. The classes of groups for which the two conjectures are knowndiffer. For example, by work of Kammeyer, Lück, and Rüping [2016] all lattice in Liegroups satisfy the Farrell-Jones Conjecture; despite Lafforgue [2002] positive results formany property T groups, the Baum-Connes Conjecture is still a challenge for SL3(Z).Wegner [2015] proved the Farrell-Jones Conjecture for all solvable groups, but the caseof amenable (or just elementary amenable) groups is open; in contrast Higson and Kas-parov [2001] proved the Baum-Connes Conjecture for all a-T-menable groups, a class ofgroups that contains all amenable groups. On the other hand, hyperbolic groups satisfyboth conjectures. See Mineyev and Yu [2002] and Lafforgue [2012] for the Baum-ConnesConjecture and, as mentioned above, A. Bartels and Lück [2012c] and A. Bartels, Lück,and Reich [2008b] for the Farrell-Jones Conjecture. For a more comprehensive summary

K-THEORY AND ACTIONS ON EUCLIDEAN RETRACTS 1061

of the current status of the Farrell-Jones Conjecture the reader is directed to Lück [2017]and Reich and Varisco [2017].

Acknowledgments. It is a pleasure to thank my teachers, coauthors, and students for themany things they taught me.

1 The Formulation of the Farrell-Jones Conjecture

Classifying spaces for families. A family F of subgroups of a group G is a non-emptycollection of subgroups that is closed under conjugation and subgroups. Examples are thefamily Fin of finite subgroups and the family VCyc of virtually cyclic subgroups (i.e., ofsubgroups containing a cyclic subgroup as a subgroup of finite index). For any family ofsubgroups of G there exists a G-CW -complex EFG with the following property: if Eis any other G-CW -complex such that all isotropy groups of E belong to F , then thereis a up to G-homotopy unique G-map E ! EFG. This space is not unique, but it isunique up to G-homotopy equivalence. Informally one may think about EFG as a spacethat encodes the group G relative to all subgroups from F . Often there are interestinggeometric models for this space, in particular for F = Fin. More information about thisspace can be found for example in Lück [2005]. An easy way to construct EFG is as theinfinite join �1i=0(

`F 2F G/F ). If F is closed under supergroups of finite index (i.e., if

F 2 F is a subgroup of finite index in F 0, then also F 0 2 F ), then the full simplicialcomplex on

`F 2F G/F is also a model for EFG; we will denote this model later by

∆F (G).

The formulation of the conjecture. The original formulation of the Farrell and Jones[1993a] used homology with coefficients in stratified and twisted Ω-spectra. Here we usethe equivalent Hambleton and Pedersen [2004] formulation developed by Davis and Lück[1998]. Given a ring R, Davis-Lück construct a homology theory X 7! HG� (X ;KR) forG-spaces with the property thatHG� (G/H ;KR) Š K�(R[H ]).

Let F be a family of subgroups of the group G. Consider the projection map EFG !G/G to the one-point G-space G/G. It induces the F -assembly map

˛GF : HG� (EF ;KR) ! H

G� (G/G;KR) Š K�(R[G]):

Conjecture 1.1 (Farrell-Jones Conjecture). For any groupG and any ringR the assemblymap ˛GVCyc is an isomorphism.

This version of the conjecture has been stated in A. Bartels, Farrell, Jones, and Reich[2004]. The original formulation of Farrell and Jones [1993a] considered only the integral

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group ring Z[G]. Moreover, Farrell and Jones wrote that they regard this and relatedconjectures only as estimates which best fit the know data at this time. However, theconjecture is still open and does still fit with all known data today.

Transitivity principle. Informally one can view the statement that the assembly map˛GF is an isomorphism for a group G and a ring R as the statement that K�(R[G]) canbe assembled from K�(R[F ]) for all F 2 F (and group homology). If V is a family ofsubgroups of G that contains all subgroups from F , then one can apply this slogan in twosteps, for G relative to V and for each V 2 V relative to the F 2 F with F � V . Theimplementation of this is the following transitivity principle.

Theorem 1.2 (Farrell and Jones [1993a] and Lück and Reich [2005]). For V 2 V setFV := fF j F 2 F ; F � V g. Assume that for all V 2 V the assembly map ˛VFV is anisomorphism. Then ˛GF is an isomorphism iff ˛

GV is an isomorphism.

Twisted coefficients. Often it is beneficial to study more flexible generalizations of Con-jecture 1.1. Such a generalization is the Fibred Isomorphism Conjecture of Farrell andJones [1993a]. An alternative is the Farrell-Jones Conjecture with coefficients in additivecategories A. Bartels and Reich [2007], here one allows additive categories with an actionof a groupG instead of just a ring as coefficients. This version of the conjecture applies inparticular to twisted group rings. These generalizations of the Conjecture have better inher-itance properties. Two of these inheritance property are stability under directed colimitsof groups, and stability under taking subgroups. For a summary of the inheritance proper-ties see Reich and Varisco [2017, Thm. 27(2)]. Often proofs of cases of the Farrell-JonesConjecture use these inheritance properties in inductions or to reduce to special cases. Wewill mean by the statement thatG satisfies the Farrell-Jones Conjecture relative to F , thatthe assembly map ˛GF is bijective for all additive categories A with G-action. However,this as a technical point that can be safely ignored for the purpose of this note.

Other theories. The Farrell-Jones Conjecture forK-theory discussed so far has an analoginL-theory, as it appears in surgery theory. For some of the applications mentioned beforethis is crucial. For example, the Borel Conjecture for a closed aspherical manifoldM ofdimensions � 5 holds if the fundamental group ofM satisfies both theK-andL-theoreticFarrell-Jones Conjecture. However, proofs of the Farrell-Jones Conjecture in K- and L-theory are by now very parallel. Recently, the techniques for the Farrell-Jones ConjectureinK- and L-theory have been extended to also cover Waldhausen’s A-theory Enkelmann,Lück, Pieper, Ullmann, and Winges [2016], Kasprowski, Ullmann, Wegner, and Winges[2018], and Ullmann and Winges [2015]. In particular, the conditions we will discuss inSection 2 are now known to imply the Farrell-Jones Conjecture in all three theories.


2 Actions on compact spaces

Amenable actions and exact groups.

Definition 2.1 (Almost invariant maps). LetX ,E beG-spaces whereE is equipped witha G-invariant metric d . We will say that a sequence of maps fn : X ! E is almostG-equivariant if for any g 2 G

supx2X

d (fn(gx); gfn(x)) ! 0 as n ! 1:

For a discrete groupG we equip the space Prob(G) of probability measures onG withthe metric it inherits as subspace of l1(G). This metric generates the topology of point-wise convergence on Prob(G). We recall the following definition.

Definition 2.2. An action of a group G on a compact space X is said to be amenable ifthere exists a sequence of almost equivariant maps X ! Prob(G).

A group is amenable iff its action on the one point space is amenable. Groups thatadmit an amenable action on a compact Hausdorff space are said to be exact or boundaryamenable. The class of exact groups contains all amenable groups, hyperbolic groupsAdams[1994], and all linear groups Guentner, Higson, and Weinberger [2005]. Other prominentgroups that are known to be exact are mapping class groups Hamenstädt [2009] and Kida[2008] and the group of outer automorphisms of free groups Bestvina, Guirardel, and Hor-bez [2017]. The Baum-Connes assembly map is split injective for all exact groups Higson[2000] and Yu [2000]. This implies the Novikov conjecture for exact groups. This is ananalytic result for the Novikov conjecture, in the sense that it has no known proof thatavoids the Baum-Connes Conjecture. There is no corresponding injectivity result for as-semblymaps in algebraicK-theory. For a survey about amenable actions and exact groupssee Ozawa [2006].

Finite asymptotic dimension. Results for assembly maps in algebraic K-theory and L-theory often depend on a finite dimensional setting; the space of probability measureshas to be replaced with a finite dimensional space. We write ∆(G) for the full simplicialcomplex with vertex set G and ∆(N )(G) for its N -skeleton. The space ∆(G) can beviewed as the space of probability measures on G with finite support. We equip ∆(G)with the l1-metric; this is the metric it inherits from Prob(G).

Definition 2.3 (N -amenable action). We will say that an action of a group G on a com-pact space X is N -amenable if there exists a sequence of almost equivariant maps X !∆(N )(G).

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The natural action of a countable groupG on its Stone-Čech compactification ˇG isN -amenable iff the asymptotic dimension ofG is at mostN Guentner, Willett, and Yu [2017,Thm. 6.5]. This condition (for any N ) also implies exactness and therefore the Novikovconjecture Higson and Roe [2000]. For groupsG of finite asymptotic dimension for whichin addition the classifying space BG can be realized as a finite CW -complex, there is analternative argument for the Novikov Conjecture Yu [1998] that has been translated tointegral injectivity results for assembly maps in algebraic K-theory and L-theory A. C.Bartels [2003] and Carlsson and Goldfarb [2004]. These injectivity results have seen farreaching generalizations to groups of finite decomposition complexity Guentner, Tessera,and Yu [2012], Kasprowski [2015], and Ramras, Tessera, and Yu [2014].

N -F -amenable actions. Constructions of transfer maps in algebraic K-theory and L-theory often depend on actions on spaces that are much nicer than ˇG. A good class ofspaces to use for the Farrell-Jones Conjecture are Euclidean retracts, i.e., compact spacesthat can be embedded as a retract in some Rn. Brouwer’s fixed point theorem implies thatfor an action of a group on an Euclidean retract any cyclic subgroup will have a fixed point.It is not difficult to check that this obstructs the existence of almost equivariant maps to∆(N ) (assuming G contains an element of infinite order). Let F be a family of subgroupsofG that is closed under taking supergroups of finite index. Let S :=

`F 2F G/F be the

set of all left cosets to members of F . Let∆F (G) be the full simplicial complex on S and∆

(N )

F (G) be its N -skeleton. We equip∆F (G) with the l1-metric.

Definition 2.4 (N -F -amenable action). We will say that an action of G on a compactspace X is N -F -amenable if there exists a sequence of almost equivariant maps X !∆

(N )

F (G). If an action is N -F -amenable for some N 2 N, then we say that it is finitelyF -amenable.

Remark 2.5. LetX be aG-CW -complexwith isotropy groups in F and of dimension� N .As∆F (G) is a model for EFG we obtain a cellular G-map f : X ! ∆

(N )

F (G); this mapis also continuous for the l1-metric. In particular, the constant sequence fn � f is almostequivariant. Therefore one can viewN -F -amenability forG-spaces as a relaxation of theproperty of being a G-CW -complex with isotropy in F and of dimension � N .

This relaxation is necessary to obtain compact examples and reasonably small F : If aG-CW -complex is compact, then it has only finitely many cells. In particular, for eachcell the isotropy group has finite index in G, so F would have to contain subgroups offinite index in G.

Theorem 2.6 (A. Bartels and Lück [2012c] and A. Bartels, Lück, and Reich [2008b]).Suppose that G admits a finitely F -amenable action on a Euclidean retract. Then Gsatisfies the Farrell-Jones Conjecture relative to F .


Remark 2.7. The proof of Theorem 2.6 depends onmethods from controlled topology/algebrathat have a long history. An introduction to controlled algebra is given in Pedersen [2000];an introduction to the proof of Theorem 2.6 can be found in A. Bartels [2016]. Here weonly sketch a very special case, where these methods are not needed.

Assume that the Euclidean retract is a G-CW -complex X . As pointed out in Re-mark 2.5 this forces F to contain subgroups of finite index in G. As X is contractible,the cellular chain complex of X provides a finite resolution C� over Z[G] of the trivialG-module Z. Note that in each degree Ck =

LZ[G/Fi ] is a finite sum of permutation

modules with Fi 2 F and Fi of finite index inG. For a finitely generated projectiveR[G]-module P we obtain a finite resolution C�˝ZP of P . Each module in the resolution isa finite sum of modules of the form Z[G/F ]˝ZP with F 2 F and of finite index in G.Here Z[G/F ]˝ZP is equipped with the diagonalG-action and can be identified with theR[G]-module obtained by first restricting P to an R[F ]-module and then inducing backup from R[F ] to R[G]. In particular,

�Z[G/F ]˝ZP

�2 K0(R[G]) is in the image of the

assembly map relative to the family F . It follows that [P ] =Pk(�1)

k [Ck˝ZP ] is alsoin the image. Therefore the assembly map H0(EFG;KR) ! K0(R[G]) is surjective.(This argument did not use that F is closed under supergroups of finite index.)

Example 2.8. LetG be a hyperbolic group. Its Rips complex can be compactified to a Eu-clidean retract Bestvina andMess [1991]. The natural action ofG on this compactificationis finitely VCyc-amenable A. Bartels, Lück, and Reich [2008a].

To obtain further examples of finitely F -amenable actions on Euclidean retracts, it ishelpful to replaceVCycwith a larger family of subgroups F . Groups that act acylindricallyhyperbolic on a tree admit finitely F -amenable actions on Euclidean retracts where F isthe family of subgroups that is generated by the virtually cyclic subgroups and the isotropygroups for the original action on the tree Knopf [2017]. Relative hyperbolic groups andmapping class groups are discussed in Section 4.

Remark 2.9. A natural question is which groups admit finitely VCyc-amenable actionson Euclidean retracts. A necessary condition for an action to be finitely VCyc-amenableis that all isotropy groups of the action are virtually cyclic. Therefore, a related questionis which groups admit actions on Euclidean retracts such that all isotropy groups are vir-tually cyclic. The only groups admitting such actions that I am aware of are hyperbolicgroups. In fact, I do not even know whether or not the group Z2 admits an action on aEuclidean retract (or on a disk) such that all isotropy groups are virtually cyclic. Thereare actions of Z2 on disks without a global fixed point. This is a consequence of Oliver’sanalysis of actions of finite groups on disks Oliver [1975]. On the other hand, there arefinitely generated groups for which all actions on Euclidean retracts have a global fixedpoint Arzhantseva, Bridson, Januszkiewicz, Leary, Minasyan, and Świ tkowski [2009].

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Homotopy actions. There is a generalization of Theorem 2.6 using homotopy actions. Inorder to be applicable to higherK-theory these actions need to be homotopy coherent. Thepassage from strict actions to homotopy actions is already visible in the work of Farrell-Jones where it corresponds to the passage from the asymptotic transfer used for negativelycurved manifolds Farrell and Jones [1986] to the focal transfer used for non-positivelycurved manifolds Farrell and Jones [1993b].

Definition 2.10 (Vogt [1973] andWegner [2012]). A homotopy coherent action of a groupG on a space X is a continuous map

Γ:

1aj=0

((G�[0; 1])j�G�X) ! X

such that

Γ(gk ; tk ; : : : ; t1; g0; x) =

8̂̂̂̂ˆ̂̂̂̂


Remark 2.13. Groups satisfying the assumptions of Theorem 2.12 are said to be homotopytransfer reducible in Enkelmann, Lück, Pieper, Ullmann, andWinges [2016]. The originalformulations of Theorems 2.6 and 2.12 were not in terms of almost equivariant maps, butin terms of certain open covers of G�X .

We recall here the formulation used for actions. (The formulation for homotopy actionsis more cumbersome.) A subsetU of aG-space is said to be an F -subset if there is F 2 Fsuch that gU = U for all g 2 F and U \ gU = ¿ for all g 2 G n F . A collection Uof subsets is said to be G-invariant if gU 2 U for all g 2 G, U 2 U. If no point iscontained in more thanN +1members of U, then U is said to be of order (or dimension)� N . A G-invariant cover by open F -subsets is said to be an F -cover.

For a compactG-spaceX we equip nowG�X with the diagonalG-action. For S � Gfinite an F -cover of G�X is said to be S -wide (in the G-direction) if

8 (g; x) 2 G�X 9U 2 U such that gS�fxg � U:

Then the action of G on X is N -F -amenable iff for any S � G finite there exists anS -wide F -cover U for G�X of dimension at most N Guentner, Willett, and Yu [2017,Prop. 4.5]. A translation from covers to maps is also used in A. Bartels and Lück [2012c],A. Bartels, Lück, and Reich [2008b], andWegner [2012]. From the point of view of covers(and because of the connection to the asymptotic dimension) it is natural to think of theNin N -F -amenable as a kind of dimension for the action of G on X , see Guentner, Willett,and Yu [2017] and Sawicki [2017].

A further difference between the formulations used above and in the references given isthat the conditions on the topology ofX are formulated differently, but certainly Euclideanretracts satisfy the condition from A. Bartels and Lück [2012c].Example 2.14. Theorem 2.12 applies to CAT(0)-groups where F = VCyc is the fam-ily of virtually cyclic subgroups A. Bartels and Lück [2012b] and Wegner [2012]. Anapplication of Theorem 2.12 to GLn(Z) will be discussed in Section 4.Remark 2.15 (The Farrell-Hsiangmethod). There are interesting groups for which one candeduce the Farrell-Jones Conjecture using Theorems 2.6 (or 2.12) and inheritance proper-ties. However, it is not clear that these methods can account for all that is currently known.A third method, going back to work of Farrell and Hsiang [1978], combines induction re-sults for finite groups Dress [1975] and Swan [1960] with controlled topology/algebra.An axiomatization of this method is given in A. Bartels and Lück [2012a]. In importantpart of the proof of the Farrell-Jones Conjecture for solvable groups Wegner [2015] is acombination of this method with Theorem 2.12.Remark 2.16 (Trace methods). TheK-theory Novikov Conjecture concerns injectivity ofassembly maps in algebraic K-theory, i.e., lower bounds for the algebraic K-theory ofgroup rings. For the integral group ring of groups, that are only required to satisfy a mild

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homological finiteness assumption, trace methods have been used by Bökstedt, Hsiang,and Madsen [1993] and Lück, Reich, Rognes, and Varisco [2017] to obtain rational injec-tivity results. The latter result in particular yields interesting lower bounds for Whiteheadgroups. For the group ring over the ring of Schatten class operators Yu [2017] proved ra-tional injectivity of the Farrell-Jones assembly map for all groups. This is the only resultI am aware of for the Farrell-Jones Conjecture that applies to all groups!

3 Flow spaces

The construction of almost equivariant maps often uses the dynamic of a flow associatedto the situation.

Definition 3.1. A flow space for a group G is a metric space FS equipped with a flow Φand an isometric G-action where the flow and the G-action commute. For ˛ > 0, ı > 0,c; c0 2 FS we write

dfol(c; c0) < (˛; ı)

to mean that there is t 2 [�˛; ˛] such that d (Φt (c); c0) < ı.

Example 3.2. Let G be the fundamental group of a Riemannian manifold M . Then thesphere bundle SM̃ equipped with the geodesic flow is a flow space for the fundamentalgroup ofM . For manifolds of negative or non-positive curvature this flow space is at theheart of the connection between K-theory and dynamics used to great effect by Farrell-Jones.

This example has generalizations to hyperbolic groups and CAT(0)-groups. For hyper-bolic groups Mineyev’s symmetric join is a flow space Mineyev [2005]. Alternatively, itis possible to use a coarse flow space for hyperbolic groups, see Remark 3.7 below. Forgroups acting on a CAT(0)-space a flow space has been constructed in A. Bartels andLück [2012b]. It consists of all parametrized geodesics in the CAT(0)-space (technicallyall generalized geodesics) and the flow acts by shifting the parametrization.

Almost equivariant maps often arise as compositions

X'�! FS

�! ∆

(N )

F (G);

where the first map is almost equivariant in an (˛; ı)-sense, and the second map is G-equivariant and contracts (˛; ı)-distances to "-distances. The following Lemma summa-rizes this strategy.

Lemma 3.3. Let X be a G-space, where G is a countable group. Let N 2 N. Assumethat there exists a flow space FS satisfying the following two conditions.


(A) For any finite subset S of G there is ˛ > 0 such that for any ı > 0 there is acontinuous map ' : X ! FS such that for x 2 X , g 2 S we have

dfol('(gx); g'(x)) < (˛; ı):

(B) For any ˛ > 0, " > 0 there are ı > 0 and a continuousG-map : FS ! ∆(N )F (G)such that

dfol(c; c0) < (˛; ı) H) d ( (c); (c0)) < "

holds for all c; c0 2 FS.

Then the action of G on X is N -F -amenable.

Proof. Let S � G be finite and " > 0. We need to construct a map f : X ! ∆(N )F (G) forwhich d (f (gx); gf (x)) < " for all x 2 X , g 2 S . Let ˛ be as in (A) with respect to S .Choose now ı > 0 and a G-map : FS ! ∆(N )F (G) as in (B). Next choose ' : X ! FSas in (A) with respect to this ı > 0. Then f := ı ' has the required property.

Remark 3.4 (On the constructions of'). Maps' : X ! FS as in condition (A) in Lemma 3.3can in negatively or non-positively curved situations often be constructed using dynamicproperties of the flow. We briefly illustrate this in a case already considered by Farrelland Jones.

Let G be the fundamental group of a closed Riemannian manifold of strict negativesectional curvatureM . Let M̃ be its universal cover and S1 the sphere at infinity for M̃ .The action of G on M̃ extends to S1. For each x 2 M̃ there is a canonical identificationbetween the unit tangent vectors at x and S1: every unit tangent vector v at x determinesa geodesic ray c starting in x, the corresponding point � 2 S1 is c(1). One say thatv points to � . The geodesic flow Φt on SM̃ has the following property. Suppose thatv and v0 are unit tangent vectors at x and x0 pointing to the same point in S1. Thendfol(Φt (v);Φt (v

0)) < (˛; ıt ) where ˛ depends only on d (x; x0) and ıt ! 0 uniformlyin v; v0 (still depending on d (x; x0)). This statement uses strict negative curvature. (Forclosed manifolds of non-positive sectional curvature the vector v0 has to be chosen morecarefully depending on v and t ; this necessitates the use of the focal transfer Farrell andJones [1993b] respectively the use of homotopy coherent actions.)

This contracting property of the geodesic flow can be translated into the constructionof maps as in (A). Fix a point x0 2 M̃ . Define '0 : S1 ! SM̃ by sending � to theunit tangent vector at x0 pointing to �. For t � 0 define 't (�) := Φt ('0(�)). Using thecontracting property of the geodesic flow is not difficult to check that for any g 2 G thereis ˛ > 0 (roughly ˛ = d (gx0; x0)) such that for any ı > 0 there is t0 satisfying

8t � t0; 8� 2 S1 dfol('t (g�); g't (�)) < (˛; ı):

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Remark 3.5. Of course the space S1 used in Remark 3.4 is not contractible and thereforenot a Euclidean retract. But the compactification M̃ [ S1 of M̃ is a disk, in particulara Euclidean retract. As M̃ has the homotopy type of a free G-CW -complex, there iseven a G-equivariant map M̃ ! ∆(d)(G) where d is the dimension ofM . In particular,the action of G on M̃ is d -amenable. It is not difficult to combine the two statementsto deduce that the action of G on M̃ [ S1 is finitely F -amenable. This is best donevia the translation to open covers of G�(M̃ [ S1) discussed in Remark 2.13, see forexample Sawicki [2017].

An important point in the formulation of condition (B) is the presence of ı > 0 uniformover FS. If the action of G on FS is cocompact, then a version of the Lebesgue Lemmaguarantees the existence of some uniform ı > 0, i.e., it suffices to construct : FS !∆

(N )

F (G) such that d ( (c); (Φt (c)) < " for all t 2 [�˛; ˛], c 2 FS.

Remark 3.6 (Long thin covers of FS). Maps ' as in condition (B) of Lemma 3.3 are bestconstructed as maps associated to long thin covers of the flow space. These long thincovers are an alternative to the long thin cell structures employed by Farrell and Jones[1986].

An open cover U of FS is said to be an ˛-long cover for FS if for each c 2 FS there isU 2 U such that

Φ[�˛;˛](c) � U:

It is said to be ˛-long and ı-thick if for each c 2 FS there is U 2 U containing the ı-neighborhood ofΦ[�˛;˛](c). The construction of mapsFS ! ∆

(N )

F (G) as in condition (B)of Lemma 3.3 amounts to finding for given ˛ an F -cover U of FS of dimension at mostN that is ˛-long and ı-thick for some ı > 0 (depending on ˛). For cocompact flowspaces such covers can be constructed in relatively great generality A. Bartels, Lück, andReich [2008a] and Kasprowski and Rüping [2017]. Cocompactness is used to guaranteeı-thickness. For not cocompact flow spaces on can still find ˛-long covers, but without auniform thickness, they do not provide the maps needed in (B).

Remark 3.7 (Coarse flow space). We outline the construction of the coarse flow spacefrom A. Bartels [2017] for a hyperbolic group G. Let Γ be a Cayley graph for G. Thevertex set of Γ is G. Adding the Gromov boundary to G we obtain the compact spaceG = G [ @G. Assume that Γ is ı-hyperbolic. The coarse flow space CF consists ofall triples (��; v; �+) with �˙ 2 G and v 2 G such that there is some geodesic from�� to �+ in Γ that passes v within distance � ı. Informally, v coarsely belongs to ageodesic from �� to �+. The coarse flow space is the disjoint union of its coarse flowlines CF��;�+ := f��g�Γ�f�+g \CF. The coarse flow lines are are quasi-isometric to R(with uniform constants depending on ı).


There are versions of the long thin covers from Remark 3.6 for CF. For ˛ > 0 these areVCyc-covers U of bounded dimension that are ˛-long in the direction of the coarse flowlines: for (��; v; �+) 2 CF there is U 2 U such that f��g�B˛(v)�f�+g \CF��;�+ � U .

There is also a coarse version of the map 't from Remark 3.4. To define it, fix a basepoint v0 2 G. For t 2 N, 't sends � 2 @G to (v0; v; �) where d (v0; v) = t and v belongsto a geodesic from v0 to � . It is convenient to extend 't to a map G�@G ! CF, with't (g; �) := (gv0; v; �) where now d (v0; v) = t and v belongs to a geodesic from gv0to �. Of course v is only coarsely well defined. Nevertheless, 't can be used to pull longthin covers for CF back to G�@G. For S � G finite there are then t > 0 and ˛ > 0such that this yields S -wide covers for G�@G. The proof of this last statement uses acompactness argument and it is important at this point that Γ is locally finite and that Gacts cocompactly on Γ.

4 Covers at infinity

The Farrell-Jones Conjecture for GLn(Z). The group GLn(Z) is not a CAT(0)-group,but it has a proper isometric action on aCAT(0)-space, the symmetric spaceX := GLn(R)/O(n).Fix a base point x0 2 X . For R � 0 let BR be the closed ball of radius R around x0.This ball is a retract of X (via the radial projection along geodesics to x0) and inherits ahomotopy coherent action ΓR from the action of GLn(Z) on X . Let F be the family ofsubgroups generated by the virtually cyclic and the proper parabolic subgroups of GLn(Z).The key step in the proof of the Farrell-Jones Conjecture for GLn(Z) in A. Bartels, Lück,Reich, and Rüping [2014] is, in the language of Section 2, the following.

Theorem4.1. The sequence of homotopy coherent actions (BR;ΓR) is finitely F -amenable.

In particular GLn(Z) satisfies the Farrell-Jones Conjecture relative to F by Theorem 2.12.Using the transitivity principle 1.2 the Farrell-Jones Conjecture for GLn(Z) can then beproven by induction on n. The induction step uses inheritance properties of the Conjectureand that virtually poly-cyclic groups satisfy the Conjecture.

The verification of Theorem 4.1 follows the general strategy of Lemma 3.3 (in a variantfor homotopy coherent actions). The additional difficulty in verifying assumption (B) isthat, as the action of GLn(Z) on the symmetric space is not cocompact, the action on theflow space is not cocompact either. The general results reviewed in Section 3 can still beused to construct for any ˛ > 0 an ˛-long cover U for the flow space. However it is notclear that the resulting cover is ı-thick, for a ı > 0 uniformly over FS. The remedy for thisshort-coming is a second collection of open subsets ofFS. Its construction starts with an F -cover for X at 1, meaning here, away from cocompact subsets. Points in the symmetricspace can be viewed as inner products on Rn and moving towards 1 corresponds todegeneration of inner products along direct summands W � Zn � Rn. This in turn

1072 ARTHUR BARTELS

can be used to define horoballs in the symmetric space, one for each W , forming thedesired cover Grayson [1984]. For each W the corresponding horoball is invariant forthe parabolic subgroup fg 2 GLn(Z) j gW = W g, more precisely, the horoballs areF -subsets, but not VCyc-subsets. The precise properties of the cover at 1 are as follows.

Lemma 4.2 (A. Bartels, Lück, Reich, and Rüping [2014] and Grayson [1984]). For any˛ > 0 there exists a collectionU1 of open F -subsets ofX of order� n that is of Lebesguenumber � ˛ at 1, i.e., there is K � X compact such that for any x 2 X n GLn(Z) �Kthere is U 2 U1 containing the ˛-ball B˛(x) in X around x.

This cover can be pulled back to the flow space where it provides a cover at 1 forthe flow space that is both (roughly) ˛-long and ˛-thick at 1. Then one is left with acocompact subset of the flow space where the cover U constructed first is ˛-long andı-thick.

This argument forGLn(Z) has been generalized toGLn(F (t)) for finite fields F , andGLn(Z[S�1]), for S a finite set of primes Rüping [2016] using suitable generalizationsof the above covers at 1. In this case the parabolic subgroups are slightly bigger, inparticular the induction step (on n) here uses that the Farrell-Jones Conjecture holds forall solvable groups. Using inheritance properties and building on these results the Farrell-Jones Conjecture has been verified for all subgroups of GLn(Q) Rüping [ibid.] and alllattices in virtually connected Lie groups Kammeyer, Lück, and Rüping [2016].

Relatively hyperbolic groups. We use Bowditch’s characterization of relatively hyper-bolic groups Bowditch [2012]. A graph is fine if there are only finitely many embeddedloops of a given length containing a given edge. Let P be a collection of subgroups ofthe countable group G. Then G is hyperbolic relative to P if G admits a cocompact ac-tion on a fine hyperbolic graph Γ such that all edge stabilizers are finite and all vertexstabilizers belong to P . The subgroups from P are said to be peripheral or parabolic. Therequirement that Γ is fine encodes Farb’s Bounded Coset Penetration property Farb [1998].Bowditch assigned a compact boundary∆ to G as follows. As a set∆ is the union of theGromov boundary @Γ with the set of all vertices of infinite valency in Γ. The topology isthe observer topology; a sequence xn converges in this topology to x if given any finite setS of vertices (not including x), for almost all n there is a geodesic from xn to x that missesS . (For general hyperbolic graphs this topology is not Hausdorff, but for fine hyperbolicgraphs it is.)

The main result from A. Bartels [2017] is that if G is hyperbolic relative to P , then Gsatisfies the Farrell-Jones Conjecture relative to the family of subgroups F generated byVCyc and P (P needs to be closed under index two supergroups here for this to includethe L-theoretic version of the Farrell-Jones Conjecture). This result is obtained as anapplication of Theorem 2.6. The key step is the following.


Theorem 4.3 (A. Bartels [ibid.]). The action of G on ∆ is finitely F -amenable.

This is a direct consequence of Propositions 4.4 and 4.5 below, using the characteriza-tion of N -F -amenability from Remark 2.13 by the existence of S -wide covers of G�∆.

To outline the construction of these covers and to prepare for the mapping class groupwe introduce some notation. Pick a G-invariant proper metric on the set E of edges of Γ;this is possible as G and E are countable and the action of G on E has finite stabilizers.For each vertex v of Γ with infinite valency let Ev be the set of edges incident to v. Writedv for the restriction of the metric to Ev . For � 2 ∆, � ¤ v we define its projection �v(�)to Ev as the set of all edges of Γ that are appear as initial edges of geodesics from v to �.This is a finite subset of Ev (this depends again of fineness of Γ). Fix a vertex v0 of finitevalence as a base point. For g 2 G, � 2 ∆ define their projection distance at v by

d�v (g; �) := dv(�v(gv0); �v(�)):

For � = v, set d�v (g; v) := 1. (For relative hyperbolic groups a related quantity is oftencalled an angle; the terminology here is chosen to align better with the case of the mappingclass group.) If we vary g (in a finite set) and � (in an open neighborhood) then for fixedv the projection distance d�v (g; �) varies by a bounded amount. Useful is the followingattraction property for projection distances: there is Θ0 such that if d�v (g; �) � Θ0, thenany geodesic from gv0 to � in Γ passes through v. Conversely, if some geodesic from gv0to � misses v, then d�v (g; �) < Θ00 for some uniform Θ00.

Projection distances are used to control the failure of Γ to be locally finite. In particular,provided all projection distances are bounded by a constantΘ, a variation of the argumentfor hyperbolic groups (using a coarse flow space), can be adapted to provide S -long coversfor the Θ-small part of G�∆. The following is a precise statement.

Proposition 4.4. There is N (depending only on G and ∆) such that for any Θ > 0 andany S � G finite there exists a collection U of openVCyc-subsets ofG�∆ that is S -wideon the Θ-small part, i.e., if (g; �) 2 G�∆ satisfies d�v (g; �) � Θ for all vertices v, thenthere is U 2 U with gS�f�g � U .

To deal with large projection distances an explicit construction can be used (similar tothe case of GLn(Z)). For (g; �) 2 G�∆ let

VΘ(g; �) := fv j d�v (g; �) � Θg:

As a consequence of the attraction property, for sufficiently largeΘ, the set VΘ(g; �) con-sists of vertices that belong to any geodesic from gv0 to �. In particular, it can be linearlyordered by distance from gv0.

For a fixed vertex v and Θ > 0 define W (v;Θ) � G�∆ as the (interior of the) setof all pairs (g; �) for which v is minimal in VΘ(g; �), i.e., v is the vertex closest to gv0

1074 ARTHUR BARTELS

for which d�v (g; �) � Θ. Then W(Θ) := fW (v;Θ) j v 2 V g is a collection of pairwisedisjoint open P -subsets of G�∆.

Proposition 4.5. Let S � G be finite. Then there are � 00 � � 0 � � � 0 such thatW(�) [ W(� 0) is a G-invariant collection of open P -subsets of order � 1 that is S -long on the � 00-large part of G�∆: if d�v (g; �) � � 00 for some vertex v, then there isW 2 W(�) [ W(� 0) such that gS�f�g � W .

A difficulty in working with the W(v;Θ) is that it is for fixedΘ not possible to controlexactly how VΘ(g; �) varies with g and � . In particular whether or not a vertex v isminimal in VΘ(g; �) can change under small variation in g or �. A consequence of theattraction property that is useful for the proof of Proposition 4.5 is the following: supposethere are vertices v0 and v1 with d�vi (g; �) > Θ

0 � Θ, then the segment between v0 and v1in the linear order of VΘ(g; �) is unchanged under suitable variations of (g; �) dependingon Θ0.Remark 4.6. Amotivating example of relatively hyperbolic groups are fundamental groupsG of complete Riemannian manifoldsM of pinched negative sectional curvature and fi-nite volume. These are hyperbolic relative to their virtually finitely generated nilpotentsubgroups Bowditch [2012] and Farb [1998]. In this case we can work with the sphereat 1 of the universal cover M̃ of M . The splitting of G�S1 into a Θ-small part and aΘ-large part can be thought of as follows. Fix a base point x0 2 M̃ . Instead of a numberΘ we choose a cocompact subset G �K of M̃ . The small part of G�∆ consists then of allpairs (g; �) for which the geodesic ray from gx0 to � is contained inX ; the large part is thecomplement. Under this translation the cover from Proposition 4.5 can again be thoughtof as a cover at 1 for M̃ . Moreover, the vertices of infinite valency in Γ correspond tohoroballs in M̃ , and projection distances to time geodesic rays spend in horoballs.

Note that the action of G on the graph Γ in the definition of relative hyperbolicity weused is cocompact, but Γ is not a proper metric space. Conversely, in the above examplethe action of G on M̃ is no longer cocompact, but now M̃ is a proper metric space. Asimilar trade off (cocompact action on non proper space versus non-cocompact action onproper space) is possible for all relatively hyperbolic groups Groves and Manning [2008],assuming the parabolic subgroups are finitely generated.

The mapping class group. Let Σ be a closed orientable surface of genus g with a finiteset P of p marked points. We will assume 6g + 2p � 6 > 0. The mapping class groupMod(Σ) of Σ is the group of components of the group of orientation preserving home-omorphisms of Σ that leave P invariant. Teichmüller space T is the space of markedcomplete hyperbolic structures of finite area on Σ n P . The mapping class group acts onTeichmüller space by changing the marking. Thurston defined an equivariant compactifi-cation of Teichmüller space T , see Fathi, Laudenbach, and Poénaru [2012]. As a space T


is a closed disk, in particular it is an Euclidean retract. The boundary of the compactifica-tion PMF := T n T is the space of projective measured foliations on Σ. The key step inthe proof of the Farrell-Jones Conjecture for Mod(Σ) is the following.

Theorem 4.7 (A. Bartels and Bestvina [2016]). Let F be the family of subgroups ofMod(Σ) that virtually fix a point in PMF . The action of Mod(Σ) on PMF is finitelyF -amenable.

From this it follows quickly that the action on T is finitely F -amenable as well, andapplying Theorem 2.6 we obtain the Farrell-Jones Conjecture for Mod(Σ) relative to F .Up to passing to finite index subgroups, the groups in F are central extensions of productsof mapping class groups of smaller complexity. Using the transitivity principle and inher-itance properties one then obtains the Farrell-Jones Conjecture for Mod(Σ) by inductionon the complexity of Σ. The only additional input in this case is that the Farrell-JonesConjecture holds for finitely generated free abelian groups.

The proof of Theorem 4.7 uses the characterization ofN -F -amenability fromRemark 2.13and provides suitable covers for Mod(Σ)�PMF . Similar to the relative hyperbolic casethe construction of these covers is done by splitting Mod(Σ)�PMF into two parts. Hereit is natural to refer to these parts as the thick part and the thin part. (The thick part corre-sponds to the Θ-small part in the relative hyperbolic case.)

Teichmüller space has a natural filtration by cocompact subsets. For " > 0 the "-thickpart T�" � T consist of all marked hyperbolic structures such that all closed geodesicshave length � ". The action of Mod(Σ) on T�" is cocompact Mumford [1971]. Fix a basepoint x0 2 T . Given a pair (g; �) 2 Mod(Σ)�PMF there is a unique Teichmüller ray cg;that starts at g(x0) and is “pointing towards �” (technically, the vertical foliation of thequadratic differential is �). The "-thick part of Mod(Σ)�PMF is defined as the set of allpairs (g; �) for which the Teichmüller ray cg;� stays in T�".

An important tool in covering both the thick and the thin part is the complex of curvesC(Σ). A celebrated result of Mazur-Minsky is that C(Σ) is hyperbolic H. A. Masur andMinsky [1999]. Klarreich [1999] studied a coarse projection map � : T ! C(Σ) andidentified the Gromov boundary @C(Σ) of the curve complex. In particular, the projectionmap has an extension � : PMF ! C(Σ) [ @C(Σ). (On the preimage of the Gromovboundary this extension is a continuous map; on the complement it is still only a coarsemap.)

Teichmüller space is not hyperbolic, but its thick part T�" has a number of hyper-bolic properties: The Masur criterion H. Masur [1992] implies that for (g; �) in the thickpart, cg;�(t) ! � as t ! 1. Moreover, the restriction of Klarreichs projection map� : PMF ! C(Σ) [ @C(Σ) to the space of all such � is injective. A result of Minsky[1996] is that geodesics c that stay in T�" are contracting. This is a property they share

1076 ARTHUR BARTELS

with geodesics in hyperbolic spaces: the nearest point projection T ! c maps balls dis-joint from c to uniformly bounded subsets. Teichmüller geodesics in the thick part T�"project to quasi-geodesics in the curve complex with constants depending only on ". Allthese properties eventually allow for the construction of suitable covers of any thick partusing a coarse flow space and methods from the hyperbolic case. A precise statement isthe following.

Proposition 4.8. There is d such that for any " > 0 and any S � Mod(Σ) finite thereexists a Mod(Σ)-invariant collection U of F -subsets of Mod(Σ)�PMF of order � dsuch for any (g; �) for which cg;� stays in T�" there is U 2 U with gS�f�g � U .

The action of the mapping class group on the curve complex C(Σ) does not exhibitthe mapping class group as a relative hyperbolic group in the sense discussed before; the1-skeleton of C(Σ) is not fine. Nevertheless, there is an important replacement for theprojections to links used in the relatively hyperbolic case, the subsurface projections ofH. A. Masur and Minsky [2000]. In this case the projections are not to links in the curvecomplex, but to curve complexes C(Y ) of subsurfaces Y of Σ. (On the other hand, oftenlinks in the curve complex are exactly curve complexes of subsurfaces.) The theory ishowever much more sophisticated than in the relatively hyperbolic case. Projections arenot always defined; sometimes the projection is to points in the boundary of C(Y ) andthe projection distance is 1. Bestvina, Bromberg, and Fujiwara [2015] used subsurfaceprojections to prove that the mapping class group has finite asymptotic dimension. Intheir work the subsurfaces ofΣ are organized in a finite numberN of familiesY such thattwo subsurfaces in the same family will always intersect in an interesting way. This hasthe effect that the projections for subsurfaces in the same family interact in a controlledway with each other. Each family Y of subsurfaces is organized in Bestvina, Bromberg,and Fujiwara [ibid.] in an associated simplicial complex, called the projection complex.The vertices of the projection complex are the subsurfaces from Y. A perturbation of theprojection distances can thought of as being measured along geodesics in the projectioncomplex and now behaves very similar as in the relative hyperbolic case, in particular theattraction property is satisfied in each projection complex. This allows the applicationof a variant of the construction from Proposition 4.5 for each projection complex thateventually yield the following.

Proposition 4.9. Let Y be any of the finitely many families of subsurfaces. For anyS � G finite there exists Θ > 0 and a Mod(Σ)-invariant collection U of F -subsetsof Mod(Σ)�PMF of order � 1 such for any (g; �) for which there is Y 2 Y withd�Y (g; �) � Θ there is U 2 U with gS�f�g � U .

The final piece, needed to combine Propositions 4.8 and 4.9 to a proof of Theorem 4.7,is a consequence of Rafi’s analysis of short curves inΣ along Teichmüller rays Rafi [2005]:


for any " > 0 there is Θ such that if some curve of Σ is "-short on cg;� (i.e., if cg;� is notcontained in T�") then there is a subsurface Y such that d�Y (g; �) � Θ.Remark 4.10. Farrell and Jones [1998] proved topological rigidity results for fundamen-tal groups of non-positively curved manifolds that are in addition A-regular. The lattercondition bounds the curvature tensor and its covariant derivatives over the manifold. Alltorsion free discrete subgroups of GLn(R) are fundamental groups of suchmanifolds. Sim-ilar to the examples discussed in this section a key difficulty in Farrell and Jones [ibid.]is that the action of the fundamental group G of the manifold on the universal cover isnot cocompact. The general strategy employed by Farrell-Jones seems however different,in particular, it does not involve an induction over some kind of complexity of G. Theonly groups that are considered in an intermediate step are polycyclic groups, and the ar-gument directly reduces from G to these and then uses computations of K-and L-theoryfor polycyclic groups.

This raises the following question: Can the family of subgroups in Theorem 4.1 be re-placed with the family of virtually polycyclic subgroups? Recall that the cover constructedfor the flow space at 1 is both ˛-long and ˛-thick, while only ı-thickness is needed. Soit is plausible that there exist thinner covers at 1 that work for the family of virtuallypolycyclic subgroups.

For the mapping class group the family from Theorem 4.7 can not be chosen to besignificantly smaller; all isotropy groups for the action have to appear in the family. Butone might ask, whether there exist N -F -amenable actions (or N -F -amenable sequencesof homotopy coherent actions) of mapping class groups on Euclidean retracts, where F issmaller than the family used in Theorem 4.7.

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Received 2017-11-29.

A BWWUMM [email protected]

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