The Farrell-Jones Conjecture for mapping class groups
aa r X i v : . [ m a t h . G T ] O c t THE FARRELL-JONES CONJECTURE FOR MAPPINGCLASS GROUPS
ARTHUR BARTELS AND MLADEN BESTVINA
Abstract.
We prove the Farrell-Jones Conjecture for mappingclass groups. The proof uses the Masur-Minsky theory of thelarge scale geometry of mapping class groups and the geometryof the thick part of Teichm¨uller space. The proof is presented inan axiomatic setup, extending the projection axioms in [14]. Morespecifically, we prove that the action of Mod(Σ) on the Thurstoncompactification of Teichm¨uller space is finitely F -amenable forthe family F consisting of virtual point stabilizers. Contents
Introduction 2Acknowledgements 111. Axioms 112. Partial covers from the projection complex 152. a . Projection covers 152. b . The projection complex and angles. 162. c . The numbers Θ , . . . , Θ . 172. d . The finite projections Z ( g, ξ ). 182. e . The open sets U ( Y, i ). 193. Partial covers from a flow space 223. a . Coarse flow spaces 233. b . Long thin covers. 243. c . The coarse flow. 253. d . Construction of U thick F -amenable actions. 294. a . Finite extensions and N - F -amenability 294. b . The Farrell-Jones Conjecture 325. Preliminaries on mapping class groups 365. a . Teichm¨uller space 36 Date : October, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Mapping Class Groups, Curve Complex, Farrell-JonesConjecture, K - and L -theory of group rings. b . Measured foliations 375. c . Measured geodesic laminations 385. d . The supporting multisurface 395. e . The Teichm¨uller metric 405. f . Holomorphic quadratic differentials 405. g . Modulus and the Collar Lemma 416. Projections 426. a . Curve complex; arc and curve complex 426. b . Curve complex of the annulus 436. c . The Gromov boundary 446. d . Subsurface projections 446. e . Projecting geodesic laminations 466. f . Projection distance 466. g . The Bounded Geodesic Image Theorem 476. h . Partitioning the subsurfaces and the color preservingsubgroup 486. i . Large intersection number implies large projection 487. Verification of the Flow Axioms 498. Teichm¨uller geodesics intersecting the thin part 528. a . Thick or thin 528. b . Outline for Proposition 8.2 538. c . Primitive annuli 549. The Farrell-Jones Conjecture for mapping class groups 57References 60 Introduction
The goal of this paper is to prove the following theorem.
Theorem A.
The mapping class group
Mod(Σ) of any oriented surface Σ of finite type satisfies the Farrell-Jones Conjecture. We will review the Farrell-Jones Conjecture and its motivation laterin the introduction. The main step in our proof of Theorem A is theverification of a regularity condition, called finite F -amenability in [4],for the action of Mod(Σ) on the space PMF of projective measuredfoliations on Σ, see Theorem B below. Theorem A is then a consequenceof the axiomatic results of L¨uck, Reich and the first author for theFarrell-Jones Conjecture from [7, 9] and an induction on the complexityof the surface. A similar induction has been used for GL n ( Z ) [11]. Topologically amenable actions.
Let G be a group. The space Prob( G )of probabilty measures on G is a subspace of ℓ ( G ). An action of G on HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 3 a compact space ∆ is said to be topologically amenable if there existsa sequence of weak ∗ continuous maps f n : ∆ → Prob( G ) satisfying thefollowing condition: for any g ∈ G sup x ∈ ∆ k f n ( gx ) − gf n ( x ) k → n → ∞ . It will be convenient to refer to sequences of maps satisfying this con-dition as almost equivariant maps . Groups that admit a topologi-cally amenable action on a compact space are often said to be bound-ary amenable . This condition is known to imply the Novikov con-jecture [40]. Since the Stone- ˘Cech compactification βG maps to anycompact G -space, it follows that a group is boundary amenable if andonly if its action on βG is topologically amenable. Hamenst¨adt provedthat the mapping class group of a surface Σ is boundary amenable byproving that its action on the space of complete geodesic laminationson Σ is topologically amenable [37]. Kida showed that the action of themapping class group on its Stone- ˘Cech compactification is topologicallyamenable [49, App. C]. Finite asymptotic dimension.
For any N the subspace Prob( G ) ( N ) ofprobability measures supported on sets of cardinality at most N + 1is naturally a simplicial complex of dimension N . The isotropy groupsfor this action all belong to the family Fin of finite subgroups. For anamenable action of G on a compact space ∆ one can ask whether thealmost equivariant maps f n from above can be chosen to have image inProb( G ) ( N ) for some N independent of n . For ∆ = βG such maps f n exist if and only if the asymptotic dimension of G is ≤ N [35, Thm. 6.5].Together with Bromberg and Fujiwara the second author proved thatthe mapping class group has finite asymptotic dimension [14]. Thisresult implies a stronger form of the Novikov conjecture [47], oftencalled the integral Novikov conjecture, i.e., the integral injectivity ofthe assembly maps of the assembly maps in algebraic K -theory and L -theory relative to the family of finite subgroups. These assembly mapsare briefly reviewed in Subsection 4. b . Finite F -amenability. The axiomatic condition from [9] that we willuse requires an action on a space much nicer than the Stone- ˘Cechcompactification. For the mapping class group this will be the Thurstoncompactification of Teichm¨uller space, i.e., a disk. Technically, therequirement is that the space is a Euclidean retract. For actions on suchspaces there are typically infinite isotropy groups and this obstructsthe existence of almost equivariant maps f n into a finite dimensionalsimplicial complex with a proper simplicial G -action. The condition ARTHUR BARTELS AND MLADEN BESTVINA from [9] is therefore formulated relative to a family F of subgroups;the action on the target simplicial complex is then allowed to haveisotropy in this family. Maps to a finite dimensional simplicial spacetranslate to finite dimensional covers of the source, as we can pull backstandard coverings of the simplicial complex. This translation is usedin the formulation of the regularity condition that we recall now indetail.Let F be a family of subgroups of a group G , i.e., F is set of subgroupsof G that is closed under conjugation and taking subgroups. A subset U of a G -space is said to be an F -subset if there is F ∈ F such that gU = U for all g ∈ F and gU ∩ U = ∅ if g F . A cover U of a G -spaceis said to be an F -cover if U is G -invariant and consists of F -subsets.If all members of U are in addition open, then we say that U is an open F -cover. For N ∈ N an action of G on a space ∆ is said to be N - F -amenable if for all finite subsets S of G there exists an open F -cover U of G × ∆ (equipped with the diagonal G -action g · ( h, x ) = ( gh, gx ))such that • the order of U is at most N ; • the cover U is S -long (in the group coordinate) , i.e., for every( g, x ) ∈ G × ∆ there is U ∈ U with gS ×{ x } ⊆ U .An action that is N - F -amenable for some N is said to be finitely F -amenable .Given such a cover one obtains almost equivariant maps f n from ∆to simplicial complexes of dimension ≤ N as follows. For each U apartition of unity subordinate to U provides a G -equivariant map f U from G × ∆ to the Nerve of U . By the first condition this nerve is ofdimension at most N . The second condition translates into the almostequivariance of the restrictions of the f U to { e }× ∆.It is straightforward to check that the action of the mapping classgroup on Teichm¨uller space is finitely Fin-amenable. The key point forus is to understand the action on the boundary of Teichm¨uller space,i.e., on the space PMF of projective measured foliations.
Theorem B.
Let Σ be a closed oriented surface of genus g with p punctures where g + 2 p − > . Let F be the family of subgroupsthat virtually fix an essential simple closed curve (up to isotopy) or arevirtually cyclic. Then the action of its mapping class group Mod(Σ) on the space
PMF of projective measured foliations on Σ is finitely F -amenable. The family F appearing in Theorem B can alternatively be describedas the family of subgroups that virtually fix a point in PMF : Firstly,
HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 5 every curve determines a point in
PMF . Secondly, every cyclic sub-group fixes a point in the Thurston compactification T of Teichm¨ullerspace by the Brouwer fixed point theorem; since the action on Te-ichm¨uller space is proper, every infinite cyclic subgroup fixes a pointin PMF . On the other hand, for any finitely F -amenable action onany space all isotropy groups of the action necessarily belong to F .Thus the family F above is, up to finite index subgroups, the smallestfamily for which the action of the mapping class group on PMF canbe finitely F -amenable. Surface groups.
To motivate our proof of Theorem B we recall themodel argument, due to Farrell-Jones [29]. We later also discuss thesituation of SL ( Z ), where the action is not cocompact. Let Σ be aclosed hyperbolic surface. Thus G = π (Σ) acts on the universal cover e Σ = H and on the circle at infinity ∆. We sketch a proof that theaction of G on ∆ is finitely F -amenable for the family F consisting ofcyclic subgroups. There are two steps in the proof, long thin covers and a geodesic flow argument .Let M = T Σ be the unit tangent bundle of Σ. This is a closed3-manifold equipped with a 1-dimensional foliation by the orbits of thegeodesic flow. Thus v, w ∈ T Σ are in the same leaf if and only if thereis a geodesic line in Σ tangent to both v and w . For any R there areonly finitely many closed leaves of length ≤ R . Step 1: Long thin covers.
For every ǫ >
R > U of M such that: • every leaf segment of length ≤ R is contained in some U ∈ U , • every U ∈ U is contained in the ǫ -neighborhood of some leaf seg-ment of length ≤ R , • the multiplicity of the cover is bounded above independently of R and ǫ .The elements of the cover are going to be small neighborhoods of theclosed leaves of length ≤ R , and otherwise they will be flow boxes forthe foliation of the form (leaf segment) × (small cross section). Caremust be taken to arrange the third bullet.Lifting this cover to T H produces an open cover e U . This is goingto be an F -cover if ǫ is sufficiently small so that π ( U ) → π ( M ) hascyclic image for every U (nontrivial for neighborhoods of closed leaves,and otherwise trivial). The number R will depend on the given finiteset S ⊂ G and then ǫ depends on R . Step 2: Geodesic flow argument.
First define the flow spaceFS = { ( x, p, ξ ) ∈ H × H × ∆ | p ∈ [ x, ξ ) } ARTHUR BARTELS AND MLADEN BESTVINA where [ x, ξ ) is the geodesic joining x to ξ . There is an embedding T H → FS onto a closed subset defined by v ( ξ − , p, ξ + ), where p ∈ H is the point where v is based, and ξ ± ∈ ∆ are the points at ±∞ of the geodesic tangent to v . The construction of the F -cover e U in the first step can also be applied to FS . (Alternatively, it suffices toextend the cover of T H such that it covers a neighborhood of T H in FS preserving the multiplicity, see Lemma 3.16). Consider for any τ ≥ ι τ : G × ∆ → FS defined as follows. First identify G with an orbit in H , thus G ⊂ H .Then let ι τ ( g, ξ ) = ( g, x τ , ξ )where x τ is the unique point on the geodesic ray [ g, ξ ) at distance τ from g . Thus ι τ is the map “flow for time τ ”. Now we argue that for τ sufficiently large the cover ι − τ ( e U ) satisfies the requirements. Thisis accomplished by a geometric limit argument: if the statement isfalse then for every τ we have ( g τ , ξ τ ) so that ι τ ( B R ( g τ ) × { ξ τ } ) isnot contained in any e U . By equivariance we may assume that themiddle coordinate of ι τ ( g τ ) belongs to a fixed compact set K (thisuses cocompactness of the action) and passing to the limit as τ → ∞ produces ( ξ − , p, ξ + ) ∈ T H not contained in any e U , contradiction. Thekey point here is that an R -ball gets squeezed by the geodesic flow toa long thin set. The group SL ( Z ) . Let G be a torsion-free subgroup of finite indexof SL ( Z ). Here we sketch a proof (close to what we do for mappingclass groups) that the standard action of G on the circle at infinity∆ is finitely F -amenable for the family F of cyclic subgroups. Theproof for surface groups breaks down since the action on H is notcocompact and it is not possible to arrange that the point ι τ ( g ) belongsto a fixed compact set. Fix an equivariant, pairwise disjoint collectionof horoballs, so that the action on the complement is cocompact andidentify G with an orbit outside the horoballs. For a fixed Θ > g, ξ ) ∈ G × ∆ is Θ-thick if the geodesic ray [ g, ξ )intersects every horoball in a segment of length ≤ Θ. Then the aboveargument generalizes to show that for any Θ > S ⊂ G thereis a required cover of the Θ-thick part of G × ∆. But we are still leftto cover the thin part.The naive idea is as follows. For every ( g, ξ ) which is not Θ-thick let B ( g, ξ ) be the first horoball that the ray [ g, ξ ) intersects in a segment HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 7 of length > Θ. Then for each horoball B define the set U ( B ) = { ( g, ξ ) | B ( g, ξ ) = B } Clearly, the collection { U ( B ) } is equivariant, covers the Θ-thin partof G × ∆, consists of F -sets, and any two are disjoint, i.e. the mul-tiplicity is 1. The problem is that the sets U ( B ) are not necessarilyopen, and the cover may not be S -long. In both cases, the issue is the threshold problem , i.e., small perturbations of ( g, ξ ) may change B ( g, ξ )dramatically.There are two things we do to solve the threshold problem. Thefirst is to modify how we measure the size of intersection between aray and a horoball. This is formalized in the notion of an angle and itsmain feature is that if a ray has long intersections with three horoballs,perturbing ( g, ξ ) does not affect the angle at the middle horoball (seeSection 2. b ). To define the angle we use the language of projectioncomplexes [14], but in the present case one could use the Farey graph,whose vertices correspond to the horoballs. This leaves the thresholdproblem only in the case of the first large intersection, and we solvethis by working at two (or technically six) different scales, see Section2. c .There is an alternative argument for SL ( Z ) where the failure ofcocompactness is addressed on the flow space [11]. This alternativeargument seems not to be applicable to the mapping class group, forexample because we have only control over the behavior of Teichm¨ullerrays that stay in a thick part. Sketch of proof of Theorem B.
Our argument follows the model argu-ments sketched above, but with several important differences. First,our geodesic flow argument, modelled on [4], is coarse , i.e. it squeezesballs to sets with bounded, rather than ǫ -small, cross-section. The flowspace FS is replaced by the coarse flow space CF .Second, instead of measuring lengths of intersections with horoballs,we use the Masur-Minsky notion of subsurface projection.The hyperbolic plane is replaced with the Teichm¨uller space T ofcomplete hyperbolic structures of finite area on Σ r P . It is equippedwith the Teichm¨uller metric, which is invariant under the action ofMod(Σ), and it is compactified by PMF . Fix a basepoint X ∈ T and identify Mod(Σ) with the orbit of X . Given a pair ( g, ξ ) ∈ Mod(Σ) ×PMF there is a unique Teichm¨uller ray c g,ξ that starts at g ( X ) and is “pointing towards ξ ” (technically, the vertical foliation ofthe quadratic differential is ξ ). The construction of the required coverof Mod(Σ) ×PMF is divided into two parts. For ǫ > g, ξ ) ARTHUR BARTELS AND MLADEN BESTVINA is ǫ -thick if no geodesic along c g,ξ has length < ǫ , and otherwise thepair is ǫ -thin.Given a finite set S ⊂ Mod(Σ) we first find ǫ > S -longcover of the ǫ -thin part. Then we cover the ǫ -thick part.When covering the thin part the main tool is the Masur-Minskynotion of subsurface projections. As in [14], the collection of all sub-surfaces is divided into finitely many subcollections Y i , i = 1 , , · · · , k so that any two subsurfaces in the same subcollection overlap. There isa finite index subgroup G <
Mod(Σ) that preserves each subcollection.Our cover of the thin part will consist of k collections of sets, one foreach Y i . Roughly speaking, a theorem of Rafi [66] guarantees that if( g, ξ ) is in the thin part then for some subsurface Y we have a largeprojection distance in Y between g ( X ) and ξ , and the elements of thecover will be parametrized by such subsurfaces. Here we use for each Y i the projection complex [14] (see also [15] for a streamlined construc-tion) to obtain a good notion of first subsurface with large projectionfor a given pair ( g, ξ ), similar to the first horoball in the model case ofSL ( Z ) above. To this end we find it useful to extend the axiomaticsetup for the projection complex to include projections of foliations ξ ∈ PMF .The strategy for covering the thick part is a coarse version of themodel arguments recalled earlier. Here we use hyperbolicity propertiesof the thick part of Teichm¨uller space. We summarized the propertieswe use in the form of flow axioms. These axioms are formulated ina quasi-isometry invariant way. We note that our fellow traveler ax-iom (F2) is weaker than what is actually known now about Teichm¨ullergeodesics in the thick part, see for example [26, § Outline by sections.
We start by listing all the axioms in Section 1. InSection 2 we construct the cover of the thin part using the projectionaxioms and in Section 3 we construct the cover of the thick part usingthe flow axioms. Section 4 contains a general discussion of the Farrell-Jones Conjecture and finite F -amenability. In Section 5 we review thebasics of mapping class groups: Teichm¨uller space, measured foliationsand geodesic laminations, Teichm¨uller metric, quadratic differentials.In Section 6 we review the Masur-Minsky subsurface projections andverify the projection axioms. In Section 7 we verify the flow axioms.The main ingredients are Minsky’s Contraction Theorem [63], the Ma-sur Criterion [59], and Klarreich’s description of the boundary of thecurve complex [50]. In Section 8 we discuss Teichm¨uller geodesics thatenter the thin part and in Section 9 we provide the roadmap showingwhich axioms are verified where. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 9
The Farrell-Jones Conjecture.
Surgery theory as developed by Brow-der-Novikov-Sullivan-Wall relates the classification of manifolds (of di-mension at least 5) to algebraic invariants, i.e., to the algebraic K -groups and L -groups of Z [ G ], the integral group ring over the fun-damental group of the manifold. Farrell and Jones [30] formulated ageneral conjecture about the structure of these K - and L -groups. Infor-mally, the conjecture asserts that the computation of K - and L -groupsof Z [ G ] can be reduced (modulo group homology) to the computationof K - and L -groups of group rings Z [ V ], where V varies over the fam-ily VCyc of virtually cyclic subgroups of G . Often it is beneficial toconsider as an intermediate step a family of subgroups F containingVCyc and to consider the Farrell-Jones Conjecture relative to F . Theformulation of the Farrell-Jones Conjecture that we will be using isrecalled in Subsection 4. b .Building on the work of Farrell and Jones their Conjecture has beenverified for many classes of groups. Among these are hyperbolic groups, CAT (0)-groups, solvable groups and lattices in Lie groups [7, 9, 45, 73,74]. A more complete list can be found in [45, Thm. 2].The Farrell-Jones Conjecture implies the Novikov Conjecture, but isstronger. Roughly speaking, the Novikov Conjecture asserts that a cer-tain map is (rationally) injective whereas the Farrell-Jones Conjectureasserts it is bijective. More concretely, as reviewed below, the Farrell-Jones Conjecture implies that aspherical manifolds are determined upto homeomorphism by their fundamental group (in dimension ≥ The surgery exact sequence.
Applications to the classification of mani-folds was the main motivation for the Conjecture and the monumentalworks surrounding it of Farrell and Jones. We give a short summary ofthe key instance of such an applications. In the following manifolds arealways topological manifolds, i.e., we do not require a smooth or PL -structure; the topological implications of the Farrell-Jones Conjectureare cleanest. Let M be a closed oriented manifold of dimension n ≥ S ( M ) of M consists of equivalenceclasses [ f : X → M ], where X is a closed topological manifold and f isa simple homotopy equivalence. We have [ f : X → M ] = [ f ′ : X ′ → M ]if and only if there is a homeomorphism h : X → X ′ with f ′ ◦ h homo-topic to f . Thus understanding the structure set amounts to classifyingall manifolds in the simple homotopy type of M . To understand thestructure set surgery theory provides the surgery exact sequence, whichwe outline next. Let L be the L -theory spectrum of the ring Z and let L h i be its connective cover. In particular, π n ( L h i ) = π n ( L ) = L n ( Z )for n ≥ π n ( L h i ) = 0 for n ≤
0. We write L s ∗ ( Z [ G ]) for the simple L -groups of the integral group ring over the fundamental group of M .The surgery exact sequence for topological manifolds can be formulatedas follows H n +1 ( M ; L ) α M −−→ L sn +1 ( Z [ G ]) → S ( M ) → H n ( M ; L h i ) α ′ M −−→ L sn ( Z [ G ]) . Here H ∗ ( − ; L h i ) and H ∗ ( − ; L ) are the homology theories associated tothe corresponding spectra, α M is Quinn’s assembly map for M and α ′ M is the composition of α M with the map induced from L h i → L . Exact-ness of this sequence includes the statement that the structure set S ( M )carries a natural group structure (which is not easily described). Thereis an extension of this sequence to a long exact sequence. The construc-tion and exactness of the surgery exact sequence for topological man-ifolds depends among other things on the work of Kirby-Siebenmannon topological manifolds and Ranicki’s identification of the geometricassembly map with the algebraic assembly map. For a more detailedsummary of these results see the introduction of [69].If G is trivial, then L s ∗ ( Z [ G ]) = L ∗ ( Z ) = H ∗ (pt; L ) and α M is themap induced by the projection M → pt for H ∗ ( − ; L ). Consequently, α M and α ′ M are both surjective. If we specialize further to M = S n ,then H n ( S n ; L h i ) ∼ = L n ( Z ) and S n ( S n ) = { id S n } , i.e., we recover thehigh-dimensional Poincar´e conjecture from the surgery exact sequence.The Farrell-Jones Conjecture gives information about the L -groupsin the surgery exact sequence. This information has the cleanest formif G is torsion free. So assume now that G satisfies the Farrell-JonesConjecture and is torsion free. Then, after some algebraic manipula-tions, there is an isomorphism α BG : H ∗ ( BG ; L ) → L sn ( Z [ G ]), see [55,Prop. 23]. Moreover, if κ : M → BG is the classifying map (inducing π ( M ) = G ), then α M = α BG ◦ κ ∗ . This yields an injection of thestructure set S ( M ) into the relative homology group H n +1 ( BG, M ; L )(where we think of M as a subspace of BG via κ ). The cokernel ofthis inclusion is a subgroup of H n ( M ; L ( Z )) = L ( Z ) = Z . Moreover,as reviewed below, the K -theory part of the Farrell-Jones conjectureimplies that the Whitehead group of G is trivial and therefore everyhomotopy equivalence X → M is simple. Consequently, the Farrell-Jones Conjecture yields an identification of all manifold structures on M with relative homology classes. Finally, let us assume in addition In fact, under our assumptions, H n +1 ( BG, M ; L ) can be identified with the ANR -homology manifold structure set and the map to Z is the Quinn obstruc-tion [69, § HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 11 that M is aspherical, i.e., that κ is a homotopy equivalence. Then ofcourse H n +1 ( BG, M ; L ) = 0 and therefore S ( M ) = { [id M ] } . In otherwords, every homotopy equivalence X → M is homotopic to a homeo-morphism, i.e., the Borel Conjecture holds for M .For the K -theory of the integral group ring Z [ G ] of a torsion freegroup the Farrell-Jones Conjecture predicts that the Loday assemblymap H n ( BG ; K Z ) → K n ( Z [ G ]) is bijective. Here H n ( − ; K Z ) is the ho-mology theory associated to the K -theory spectrum K Z of the integers Z . Since K n ( Z ) = 0 for n < K ( Z ) = Z , K ( Z ) = Z × ∼ = Z / Z , theFarrell-Jones conjecture predicts for torsion free groups K n ( Z [ G ]) = 0for n < K ( Z [ G ]) = Z and K ( Z [ G ]) = Z / Z × G ab ; in particular, itpredicts the vanishing of the reduced projective class group ˜ K ( Z [ G ])and of the Whitehead group Wh( G ) [55, Cor. 67]. The Farrell-JonesConjecture has applications to a number of further conjectures aboutgroup rings, for example Kaplansky’s idempotent conjecture, Bass’Conjecture about the Hattori-Stallings rank, and Serre’s Conjecturethat groups of type FP are of type FF [10, 54, 55]. Acknowledgements.
Our collaboration started during a workshopat the Hausdorff Institute for Mathematics in Bonn in April 2015. Wethank Wolfgang L¨uck for inviting us to this workshop.We thank Ken Bromberg, Jon Chaika, Howard Masur, and KasraRafi for many interesting conversations about Teichm¨uller theory. Weespecially thank Kasra Rafi for his help with Theorem 8.1.We thank the referee for a long list with helpful comments.The first author is supported by the SFB 878 in M¨unster. The secondauthor is supported by the NSF under the grant number DMS-1308178.1.
Axioms
Let G be a group, and let ∆ be a G -space. We assume that ∆ iscompact, metrizable and finite dimensional. In this section we discusssomewhat elaborate axioms that will imply that the action of G on ∆is finitely F -amenable. The axioms will be defined in the presence oftwo pieces of further data. Definition 1.1.
Projection data for the action of G on ∆ consists of • a finite collection of G -sets Y = { Y , Y , · · · , Y k } ; • for each Y ∈ Y and each Y ∈ Y an open subspace ∆( Y ) ⊆ ∆ anda map d πY : (cid:0) Y r { Y } (cid:1) × (cid:0) ∆( Y ) ∐ Y r { Y } (cid:1) → [0 , ∞ ] . We also require G -equivariance, i.e., for g ∈ G , Y ∈ Y , we require∆( gY ) = g ∆( Y ) and for X ∈ Y r { Y } , ξ ∈ ∆( Y ) ∐ Y r { Y } werequire d πgY ( gX, gξ ) = d πY ( X, ξ ). We also require d πY ( X, Z ) < ∞ for X, Y, Z ∈ Y . We will refer to the d πY as projection distances . Definition 1.2.
Flow data for the action of G on ∆ consists of • a proper metric space T with a proper isometric G -action; • a compact metrizable space T = T ⊔ ∆ such that T ⊆ T is openand the G -actions on T and ∆ combine to a continuous G -actionon T ; • for every compact subset K ⊆ T a collection G K of ( µ, A K )-quasi-geodesic rays c : [0 , ∞ ) → G · K ⊆ T .The space T is assumed to be finite dimensional, separable and metriz-able. As indicated with the notation, the additive constant A K forquasi-geodesic rays is uniform for all c ∈ G K , but is allowed to dependon K , while the multiplicative constant µ is required to be independentof K as well. In our application we will have ( µ, A K ) = (1 , c ∈ G K will be a geodesic ray. We also require that G K is G -invariant.Finally, we require that each ray c ∈ G K has a limit c ( t ) → c ( ∞ ) ∈ ∆. Remark . Examples will follow the axioms, but we point out thefollowing. First, the projection distance d πY ( X, Z ) will always be finitewhen
X, Y, Z are distinct elements of Y . On the other hand, d πY ( X, ξ )may be infinite or undefined when ξ ∈ ∆. Second, the collection G K should be thought of as really depending only on G · K and consists of(quasi)geodesic rays in T that are contained in G · K , which is thoughtof as the “thick part”.We now fix base points X Y ∈ Y and a base point x ∈ T . The choiceof the base points only affects the values of Θ and K later on, but notthe validity of the axioms. Axiom 1.4 (Projections) . For each Y ∈ Y the projection distances( d πY ) Y ∈ Y satisfy the following projection axioms with respect to someconstant θ ≥ Symmetry.
For
X, Z ∈ Y r { Y } , d πY ( X, Z ) = d πY ( Z, X ) . (P2) Triangle inequality.
For all
X, Z ∈ Y r { Y } , ξ ∈ ∆( Y ) ∐ Y r { Y } d πY ( X, Z ) + d πY ( Z, ξ ) ≥ d πY ( X, ξ ) . (P3) Inequality on triples.
For all ξ ∈ ∆( Y ) ∩ ∆( Y ′ ) ∐ Y r { Y, Y ′ } , Y = Y ′ we havemin { d πY ( Y ′ , ξ ) , d πY ′ ( Y, ξ ) } < θ. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 13 (P4)
Finiteness.
For all
X, Z ∈ Y the set (cid:8) Y ∈ Y \ { X, Z } | d πY ( X, Z ) > θ (cid:9) is finite.(P5) Coarse semi-continuity.
For ξ ∈ ∆( Y ), X ∈ Y \ { Y } , θ < Θ < ∞ with d Y ( X, ξ ) ≥ Θ there exists a neighborhood U of ξ in ∆such that U ⊆ ∆( Y ) and for all ξ ′ ∈ U , d πY ( X, ξ ′ ) > Θ − θ. Axiom 1.5 (Thick or thin) . For every Θ > K ⊆ T compactsuch that for any ( g, ξ ) ∈ G × ∆(a) either, there is c ∈ G K with c (0) = gx and c ( ∞ ) = ξ (b) or, there are Y ∈ Y , Y ∈ Y with ξ ∈ ∆( Y ) and d πY ( gX Y , ξ ) > Θ.Pairs ( g, ξ ) to which (a) of Axiom 1.5 applies will be said to belongto the K -thick part of G × ∆. Pairs ( g, ξ ) to which (b) of Axiom 1.5applies will be said to admit a Θ -large projection . Axiom 1.6 (Flow axioms) . Let K ⊆ T be compact.(F1) Small at ∞ . Let c n ∈ G K , x n ∈ T such that d T (Im( c n ) , x ) and d T ( c n (0) , x n )are bounded. Then x n → ξ ∈ ∆, if and only if c n (0) → ξ .(F2) Fellow traveling.
For any ρ >
R > x ∈ T , all ξ + ∈ ∆, all t ∈ [0 , ∞ ) there exists anopen neighborhood U + of ξ + in ∆ with the following property.Let c, c ′ ∈ G K be two quasi-geodesic rays that both start in the ρ -neighborhood of x and satisfy c ( ∞ ) , c ′ ( ∞ ) ∈ U + . We requirethat d T ( c ( t ) , c ′ ( t )) < R .(F3) Infinite quasi-geodesics.
For all ρ >
R > ξ − , ξ + ∈ ∆ we define T K,ρ ( ξ − , ξ + ) ⊆ T to consist of all x for which there are c n ∈ G K with c n (0) → ξ − , c n ( ∞ ) → ξ + and d T (Im( c n ) , x ) ≤ ρ . If T K,ρ ( ξ − , ξ + ) = ∅ ,then we require that there exists a quasi-geodesic c : R → T such that T K,ρ ( ξ − , ξ + ) is contained in the R -neighborhood of c .Here the additive constant for c depends only on K , while themultiplicative constant for c is required to not depend on anychoices. Remark . The fellow traveling axiom (F2) implies that for any α >
R > c, c ′ ∈ G K with d T ( c (0) , c ′ (0)) ≤ α and c ( ∞ ) = c ′ ( ∞ ) we have ∀ t ≥ d T ( c ( t ) , c ′ ( t )) ≤ R. Remark . If T is a proper δ -hyperbolic space and ∆ is its Gromovboundary, then (F1)-(F3) hold, with G K consisting of all geodesicscontained in the “thick part” G · K . Moreover, (F1)-(F2) hold when T is a proper CAT (0) space and ∆ its visual boundary. However,(F3) may fail, e.g. when T = R and ξ ± are two antipodal points onthe boundary circle. It is possible to weaken (F3) and only demandthat T K,ρ ( ξ − , ξ + ) has a doubling property, see Proposition 3.5. Thecollection F in Theorem 1.11 below would have to be suitably enlargedto include stabilizers of such sets. Example . To fix ideas we point out that the axioms hold for thegroup G = SL ( Z ) acting on the hyperbolic plane T = H with itsGromov boundary ∆ = S . Fix a pairwise disjoint and equivariantcollection Y of open horoballs. Projection data Y will consists of thesingle collection Y , and we will set ∆( Y ) = ∆ for all Y ∈ Y . If Y, Z are distinct horoballs, denote by π Y ( Z ) the nearest point projectionof Z to the closure of Y . This is an open interval in the horocycleboundary of Y and it has uniformly bounded diameter. If X, Y, Z aredistinct horoballs we set d πY ( X, Z ) = diam (cid:0) π Y ( X ) ∪ π Y ( Z ) (cid:1) where the diameter is taken with respect to the metric in T (or if onewishes in the path metric of the horocycle; they are coarsely equivalentand the distinction is irrelevant). Similarly, if ξ ∈ ∆ is a point on thecircle that does not belong to the closure of the horoball Y , there is awell defined nearest point projection π Y ( ξ ) in the boundary of Y , andwe again define d πY ( X, ξ ) = diam (cid:0) π Y ( X ) ∪ π Y ( ξ ) (cid:1) Finally, if ξ is in the closure of Y , we define d πY ( X, ξ ) = ∞ for allhoroballs X = Y . Here π Y ( ξ ) is not defined, but we may think of it asthe point at infinity of the horocycle.To complete the description of the flow data, we define G K as theset of geodesic rays contained in G · K , for any compact K ⊂ T .Verification of the axioms is left to the reader. Example . Let G be a hyperbolic group relative to a finitely gen-erated subgroup H (or more generally relative to a collection of suchsubgroups). Then G acts on a proper δ -hyperbolic space (see [22, 27,33, 34]) with Gromov boundary ∆ and the action is cocompact in thecomplement of a pairwise disjoint equivariant collection of horoballs.Projection and flow data can be constructed in the same way as aboveand all axioms hold. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 15
The Farrell-Jones conjecture for relatively hyperbolic groups wasproved in [4] and generalizing this argument to mapping class groupswas the motivation for the present work.For the description of flow and projection data in the case of mappingclass groups, see Section 9.
Theorem 1.11.
Assume that there is projection data and flow datafor the action of G on ∆ satisfying the axioms from 1.4, 1.5 and 1.6.Assume that ∆ is compact, metrizable and finite dimensional. If, inaddition, F contains the family VCyc of virtually cyclic subgroups of G and all isotropy groups for the action of G on all the Y ∈ Y , thenthe action of G on ∆ is finitely F -amenable.Proof. This will be an easy consequence of Theorems 2.1 and 3.1 thatwe prove later.Let S ⊆ G finite. Let us say that a collection of open subsets of G × ∆is S -long at ( g, ξ ) if for one of its members U we have gS ×{ ξ } ⊆ U .Theorem 2.1 provides for each Y ∈ Y a G -invariant collection U Y ofopen F -subsets and a number Θ such that U thin := S Y ∈Y U Y is S -longat all ( g, ξ ) that admit a Θ-large projection. The order of each U Y isat most 1 and therefore the order of U thin is at most N thin := 2 k − K ⊆ G compact such thatall ( g, ξ ) that do not admit a Θ-large projection belong to the K -thickpart of G × ∆.Theorem 3.1 provides a G -invariant collection U thick of open VCyc-subsets of G × ∆ that is S -long at all ( g, ξ ) from the K -thick part.Moreover, the order of U thick is bounded by a number N thick independentfrom K and S .Altogether U thin ∪ U thick is the cover we need and the action of G on∆ is ( N thin + N thick + 1)- F -amenable. (cid:3) Partial covers from the projection complex a . Projection covers.
Throughout this section we consider a G -space ∆. We assume that we are given a G -set Y , subsets ∆( Y ) ⊆ ∆for Y ∈ Y and projection distances d πY : (cid:0) Y r { Y } (cid:1) × (cid:0) ∆( Y ) ∐ Y r { Y } (cid:1) → [0 , ∞ ] . For ξ ∈ ∆, X ∈ Y we set d π Y ( X, ξ ) := sup { d πY ( X, ξ ) | ξ ∈ ∆( Y ) } . Ifthere is no such Y , then we set d π Y ( X, ξ ) = −∞ . We also write F Y forthe family of subgroups of G that fix an element of Y . Theorem 2.1.
Assume that the projection distances ( d πY ) Y ∈ Y satisfythe axioms (P1) to (P5) listed in 1.4. Pick a base point X Y ∈ Y . Let S ⊆ G be finite. Then there is Θ ≥ and a G -invariant collection U of open F Y -subsets of G × ∆ such that(a) the order of U is at most ;(b) for any ( g, ξ ) ∈ ( G × ∆) with d π Y ( gX Y , ξ ) ≥ Θ there is U ∈ U with gS ×{ ξ } ⊆ U . The proof of Theorem 2.1 occupies the remainder of this section.2. b . The projection complex and angles.
The restrictions of the d πY to Y r { Y } satisfy the projection axioms (PC1) to (PC4) from [14,Sec. 3.1]. These axioms allow, for a constant K >> θ , the constructionof the projection complex P K ( Y ) [14, Sec. 3.3]. This complex is a graphwhose set of vertices is Y . We write d P for the path metric with edgesof length 1 on P K ( Y ). The action of G on Y extends to a simplicialaction on P K ( Y ). Crucial for us will be the following two propertiesfrom [14] of the projection complex. Proposition 2.2.
There is a constant θ ′P > (depending on K ) suchthat the following holds.(a) Local estimate. If Y is an internal vertex of a geodesic in P K ( Y ) from X to Z , and if X ′ and Z ′ are two vertices on this geodesicsuch that X ′ is between X and Y , and Z ′ is between Y and Z then | d πY ( X, Z ) − d πY ( X ′ , Z ′ ) | < θ ′P .(b) Attraction property. If d πY ( X, Z ) > θ ′P , then any geodesic in P K ( Y ) between X and Z will pass through Y .Proof. The construction of P K ( Y ) depends on a bounded perturbation d Y of the restrictions of the d πY to Y r { Y } , i.e., | d πY ( X, Z ) − d Y ( X, Z ) | is uniformly bounded in X, Y and Z , see [14, Thm. 3.3(B)]. Let c bea geodesic in P K ( Y ) from X to Y and let X ′ be any internal vertexof c . Then d Y ( X, X ′ ) is uniformly bounded by [14, Cor. 3.15]. Thus d πY ( X, X ′ ) is also uniformly bounded and the local estimate followsfrom the triangle inequality.For d Y the attraction property is the content of the first statementof [14, Lem. 3.18]. Since d Y is a uniformly bounded perturbation of d πY , the attraction property follows for d πY as well. (cid:3) For an internal vertex Y of a geodesic c in the projection complexwe define the angle of c at Y as (cid:30) Y c := d πY ( X, Z ) where X and Z arethe two vertices on c adjacent to Y . If Y is disjoint from c , then weset (cid:30) Y c = 0; if Y is the start or end point of c , then (cid:30) Y c remainsundefined. For X, Z = Y we set d max Y ( X, Z ) := max { (cid:30) Y c } , where c varies over the set of all geodesics from X to Z . HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 17
We record the following consequences of Proposition 2.2 for thesethe definitions. Item (d) is the main advantage d max has over d π . Lemma 2.3.
There is θ P > such that for all vertices X, Z = Y (a) | d πY ( X, Z ) − d max Y ( X, Z ) | < θ P ;(b) for any geodesic c from X to Z we have d max Y ( X, Z ) − (cid:30) Y c < θ P ;(c) if d max Y ( X, Z ) ≥ θ P or d πY ( X, Z ) ≥ θ P then any geodesic from X to Z passes through Y ;(d) let c be a geodesic from X to Z that passes through Y and Y inthis order, if max { d πY ( X, Y ) , d πY ( X, Z ) , (cid:30) Y c } ≥ θ P , then d max Y ( X, Z ) = d max Y ( Y , Z ) . Note that there is a uniform bound on the difference between any ofthe three numbers appearing in the hypothesis of item (d) in the abovelemma.
Proof of Lemma 2.3.
For θ P > θ ′P , properties (a) and (b) are a con-sequence of the local estimate provided there exists a geodesic from X to Z that passes through Y . If there is no such geodesic, properties (a)and (b) follow from the attraction property. For θ P > θ ′P , property (c)follows again from the attraction property. Property (c) implies thatunder the assumption of (d) any geodesic from X to Z can be built byconcatenation of a geodesic from X to Y with a geodesic from Y to Z . Thus (d) holds. (cid:3) c . The numbers Θ , . . . , Θ .Lemma 2.4. For S ⊂ Y finite, there is θ ′ S ≥ such that for all X, X ′ ∈ S , Y, Z ∈ Y , | d πY ( X, Z ) − d πY ( X ′ , Z ) | < θ ′ S . Proof.
By finiteness (P4), d πY ( X, X ′ ) is bounded for Y ∈ Y , X, X ′ ∈ S . (cid:3) Lemma 2.5.
For S ⊂ Y finite, there is θ S ≥ such that for X ∈ S , Y, Z ∈ Y , X, Z = Y we have(a) the distance between any two numbers from { d πY ( X ′ , Z ) , d max Y ( X ′ , Z ) | X ′ ∈ S } is < θ S ;(b) if d max Y ( X, Z ) ≥ θ S or d πY ( X, Z ) ≥ θ S then for any X ′ ∈ S anygeodesic from X ′ to Z passes through Y ; (c) suppose there is a geodesic c from X to Z that passes through Y and Y in this order, if one of the numbers d πY ( X, Y ) , d max Y ( X, Y ) , d πY ( X, Z ) , d max Y ( X, Z ) is ≥ θ S , then for any X ′ ∈ S d max Y ( X ′ , Z ) = d max Y ( Y , Z ) .Proof. This follows by combining Lemma 2.4 with Lemma 2.3. (cid:3)
Fix now S ⊆ G finite. Let θ S := θ S · X Y as in Lemma 2.5. Nextchoose numbers 0 << Θ << Θ << Θ << Θ << Θ << Θ . Laterwe will need estimates of the form Θ i > Θ j + C for i > j and for C aconstant depending on θ and θ S and it will be clear that we can choosethe Θ i at this point to satisfy all required estimates. (On the otherhand, Θ i := 10 · ( i + 1) · ( θ + θ S ) will certainly work.)2. d . The finite projections Z ( g, ξ ) . For all ( g, ξ ) ∈ G × ∆ with d π Y ( gX Y , ξ ) > Θ we pick Z ( g, ξ ) ∈ Y such that d πZ ( g,ξ ) ( gX Y , ξ ) > Θ .In addition, if possible, choose Z ( g, ξ ) so that d πZ ( g,ξ ) ( gX Y , ξ ) > Θ . Wecan arrange this map to be G -equivariant, i.e., such that Z ( hg, hξ ) = hZ ( g, ξ ) for h ∈ G . Remark . The use of the axiom of choice to produce the Z ( g, ξ ) ∈ Y may seem a little heavy handed. Assuming that Y is countable we cando this, with a little more care, using only countable choice:By equivariance it suffices to choose the Z (1 , ξ ) with 1 ∈ G the unit.Fix a countable basis U i of open sets for ∆ and consider all pairs ( U i , Y )such that d πY ( X Y , ξ ) > Θ for all ξ ∈ U i . Choose an ordering of thiscountable set of pairs. Then for ξ let Z (1 , ξ ) = Y where ( U i , Y ) is thefirst pair with ξ ∈ U i . By the coarse semi-continuity axiom (P5) thisproduces Z (1 , ξ ) for all (1 , ξ ) with d π Y (1 , ξ ) > Θ + θ . It is not difficultto adjust the constants in the rest of our argument to account for thisslightly weaker statement. Lemma 2.7.
Let ( g, ξ ) ∈ G × ∆ with d π Y ( gX Y , ξ ) > Θ . Then there isan open neighborhood U of ξ in ∆ and Y ∈ Y such that for any s ∈ S , ξ ′ ∈ U either Z ( gs, ξ ′ ) = Y or d max Y ( gsX Y , Z ( gs, ξ )) > Θ .Proof. It suffices to consider g = e . By coarse semi-continuity (P5) wefind a neighborhood U of ξ in ∆ such that d πZ ( e,ξ ) ( X Y , ζ ) > Θ − θ forall ζ ∈ U . For s ∈ S then d πZ ( e,ξ ) ( sX Y , ζ ) > Θ − θ S − θ > Θ byLemma 2.5 (a). In particular, for all s ∈ S , ζ ∈ U , the vertex Z ( s, ζ )is defined. We claim that for ( s, ζ ) ∈ S × U with Z ( s, ζ ) = Z ( e, ξ )(2.8) max { d πZ ( s,ζ ) ( X Y , Z ( e, ξ )) , d πZ ( e,ξ ) ( X Y , Z ( s, ζ )) } > Θ + θ S . HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 19
Indeed, assume d πZ ( e,ξ ) ( X Y , Z ( s, ζ )) ≤ Θ + θ S . Then d πZ ( e,ξ ) ( Z ( s, ζ ) , ζ ) ≥ d πZ ( e,ξ ) ( X Y , ζ ) − d πZ ( e,ξ ) ( X Y , Z ( s, ζ )) > Θ − Θ − θ S > θ. The inequality on triples implies d πZ ( s,ζ ) ( Z ( e, ξ ) , ζ ) < θ . Thus d πZ ( s,ζ ) ( X Y , Z ( e, ξ )) >d πZ ( s,ζ ) ( sX Y , ζ ) − d πZ ( s,ζ ) ( X Y , sX Y ) − d πZ ( s,ζ ) ( Z ( e, ξ ) , ζ ) > Θ − θ S − θ > Θ + θ S , proving (2.8). Now we combine (2.8) with Lemma 2.5 (b), (c). Thusfor ( s, ζ ) ∈ S × U with Z ( s, ζ ) = Z ( e, ξ ) we have • either, for every s ′ ∈ S , every geodesic from s ′ X Y to Z ( e, ξ ) passesthrough Z ( s, ζ ) and d max Z ( s,ζ ) ( s ′ X Y , Z ( e, ξ )) ≥ Θ , • or, for every s ′ ∈ S , every geodesic from s ′ X Y to Z ( s, ζ ) passesthrough Z ( e, ξ ) and d max Z ( e,ξ ) ( s ′ X Y , Z ( s, ζ )) ≥ Θ .If there is no ( s, ζ ) to which the first case applies, then we set Y := Z ( e, ξ ). Otherwise, we pick Y among the Z ( s, ζ ) to which the first caseapplies of minimal distance to X Y . Since this Y is also among thosethe Z ( s, ζ ) of maximal distance from Z ( e, ξ ), it follows that it is alsoof minimal distance from s ′ X Y for all s ′ ∈ S . Now, Lemma 2.4 (c)implies that whenever Z ( s, ζ ) = Y , then d max Y ( sX Y , Z ( s, ζ )) ≥ Θ . (cid:3) Remark . A key tool from [14] are linear orders constructed fromthe projection distances. For
X, Z ∈ Y there is a linear order on theset of all Y ∈ Y for which d πY ( X, Z ) is defined and large. It is naturalto add X as a minimal element and Z as a maximal element to thislinear order; we will then call it the linear order from X to Z . In thisorder Y < Y ′ if and only if d πY ( X, Y ′ ) is large. Using these orders theconstruction of Y in Lemma 2.7 can be summarized as follows: Foreach pair ( s, ζ ) either Z ( s, ζ ) belongs to the linear order from X Y to Z ( e, ξ ) or Z ( e, ξ ) belongs to the order from X Y to Z ( s, ζ ). The possiblepositions of Z ( s, ζ ) in these orders are sketched in Figure 1. Then Y can be defined as the minimal element in the order from X Y to Z ( e, ξ )among all Z ( s, ζ ) to which the first case applies. This is well definedas these linear orders are finite by the finiteness axiom (P4).2. e . The open sets U ( Y, i ) . For ( g, ξ ) ∈ G × ∆ with d π Y ( gX Y , ξ ) > Θ we now use the projection complex P K ( Y ) to make the followingdefinitions. • For i = 0 , , Y i ( g, ξ ) of P K as the unique vertexwith the following two properties. Firstly, d max Y ( gX Y , Y i ( g, ξ )) < Θ i X Y ξs ′ X Y sX Y ζ ζ ′ Z ( s, ζ ) Z ( e, ξ ) Z ( s ′ , ζ ′ ) Figure 1.
Possible positions of Z ( s, ζ )for all Y ∈ Y r { gX Y , Y i ( g, ξ ) } . Secondly, Y i ( g, ξ ) = Z ( g, ξ ) or thereexists a geodesic c from gX Y to Z ( g, ξ ) with d max Y i ( g,ξ ) ( gX Y , Z ( g, ξ )) ≥ Θ i . (The uniqueness of Y i ( g, ξ ) is a consequence of Lemma 2.3 sinceΘ i > θ P .) • For Y ∈ Y , i = 1 , U + ( Y, i ) ⊆ G × ∆ to consist of all( g, ξ ) with d π Y ( gX Y , ξ ) > Θ and Y = Y i ( g, ξ ). We define U ( Y, i )as the interior of U + ( Y, i ) in G × ∆. For i = 1 , U ( i ) := { U ( Y, i ) | Y ∈ Y } . Lemma 2.10.
For Y = Y ′ ∈ Y we have U ( Y, i ) ∩ U ( Y ′ , i ) = ∅ . For g ∈ G , Y ∈ Y we have g ( U ( Y, i )) = U ( gY, i ) .Proof. This is a direct consequence of the definition of U ( Y, i ). (cid:3) Lemma 2.11.
Let ( g, ξ ) ∈ G × ∆ with d π ( gX Y , ξ ) > Θ . Then thereare Y ∈ Y and i ∈ { , } with gS ×{ ξ } ⊆ U ( Y, i ) .Proof. We can assume g = e . Set Y i := Y i ( e, ξ ) for i = 0 , , Y ∈ Y and an open neighbor-hood U of ξ in ∆ such that for any s ∈ S , ζ ∈ U , either Y = Z ( s, ζ ) or d max Y ( sX Y , Z ( s, ζ )) > Θ . This implies, since Θ > θ P , byLemma 2.3 (d),(2.12) d max Y ( sX Y , Z ( s, ζ )) = d max Y ( sX Y , Y )for any Y on a geodesic from sX Y to Y .We claim that for any internal vertex Y of a geodesic from Y to Y and any s ∈ S , ζ ∈ U we have(2.13) d max Y ( sX Y , Z ( s, ζ )) = d max Y ( X Y , Z ( e, ξ )) . To prove this claim we observe first d max Y ( X Y , Y ) > d max Y ( X Y , Z ( e, ξ )) − θ S ≥ Θ − θ S > θ S . Therefore Lemma 2.5 (c) implies d max Y ( sX Y , Y ) = d max Y ( X Y , Y ) for all s ∈ S . Now (2.13) follows from (2.12). HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 21
Next we claim that, provided Y = Y , we have(2.14) | d max Y ( sX Y , Z ( s, ζ )) − d max Y ( X Y , Z ( e, ξ )) | < θ S for all s ∈ S , ζ ∈ U . To prove this claim we note that by (2.12)to we have, d max Y ( X Y , Y ) = d max Y ( X Y , Z ( e, ξ )) ≥ Θ > θ S . Thus, byLemma 2.5 (b), any geodesic from sX Y to Y will pass through Y .Using again (2.12) we have d max Y ( sX Y , Z ( s, ζ )) = d max Y ( sX Y , Y ) for s ∈ S , ζ ∈ U . Now Lemma 2.5 (a) implies (2.14).Let for s ∈ S , Y be an internal vertex of a geodesic from sX Y to Y .We claim that then for any ζ ∈ U (2.15) d max Y ( sX Y , Z ( s, ζ )) < Θ . Suppose, by contradiction, d max Y ( sX Y , Z ( s, ζ )) ≥ Θ . Then, by (2.12), d max Y ( sX Y , Y ) ≥ Θ . Lemma 2.5 (a) implies d max Y ( X Y , Y ) > Θ − θ S > Θ . Using (2.12) again, we have d max Y ( X Y , Z ( e, ξ )) > Θ . By definitionof Y and Y , this implies that Y is closer to Y then Y . But thiscontradicts that Y is closer to sX Y than Y . This establishes (2.15).Now, if Y = Y = Y = Y , then, by (2.14), d max Y ( sX Y , Z ( s, ζ )) ≥ d max Y ( X Y , Z ( e, ξ )) − θ S ≥ Θ − θ S > Θ . Using (2.15) this implies Y = Y = Y ( s, ζ ) for all s ∈ S , ζ ∈ U . Thus,in this case, S × U ⊆ U + ( Y ,
1) and therefore S ×{ ξ } ⊆ U ( Y , Y = Y = Y then (2.13), (2.14) and (2.15), imply Y ( s, ζ ) = Y for all s ∈ S , ζ ∈ U . Thus, in this case S × U ⊆ U + ( Y ,
2) and therefore S ×{ ξ } ⊆ U ( Y , Y = Y , then we use in addition that d max Y ( sX Y , Z ( s, ζ )) > Θ for all s ∈ S , ζ ∈ U with Z ( s, ζ ) = Y . Combining this with (2.13),(2.14) and (2.15), we find again Y ( s, ζ ) = Y for all s ∈ S , ζ ∈ U . Thus,also in this case S × U ⊆ U + ( Y ,
2) and therefore S ×{ ξ } ⊆ U ( Y , (cid:3) Remark . Informally the key observation in the proof of Lemma 2.11is: angles at Y and Y depend on ( s, ζ ) only up to a bounded errorand all other angles behave as indicated in Figure 2. X Y ξsX Y ζY Y Y Y angles are small angles do notdepend on ( s, ζ ) Figure 2.
The position of the Y i Conclusion of Proof of Theorem 2.1.
We can use U := U (1) ∪ U (2) andΘ := Θ . Lemma 2.10 implies that the order of U is at most 1 and thatits members are F Y -sets. Lemma 2.11 states that U has the propertyrequired in (b) in the G -direction. (cid:3) Partial covers from a flow space
Throughout this section ∆ will be a finite dimensional, metrizable,compact space with a G -action. Moreover, T = T ∪ ∆, G K will be flowdata as in Definition 1.2.We fix a base point x ∈ T and define for K ⊆ T compact( G × ∆) K ⊆ G × ∆to consist of all ( g, ξ ) for which there exists a ray c ∈ G K with c (0) = gx and c ( ∞ ) = ξ , i.e., ( G × ∆) K is the K -thick part of G × ∆. Theorem 3.1.
Assume that the flow axioms (F1), (F2) and (F3) listedin 1.6 are satisfied. Then there exists a number N thick with the follow-ing property. For S ⊆ G finite and K ⊆ T compact, there exists a G -invariant collection U thick of open F -subsets of G × ∆ such that thefollowing two conditions are satisfied:(a) the order of U thick is at most N thick ;(b) for any ( g, ξ ) ∈ ( G × ∆) K there is U ∈ U thick with gS ×{ ξ } ⊆ U . The proof of this result is in two steps. We first build a coarseflow space that admits long thin covers, i.e., covers that have a largeLebesgue number in the direction of the flow. Then we coarsely map( G × ∆) K to this flow space and show that for large time the coarseflow sends gS ×{ ξ } into such long and thin sets. The cover U thick isthen obtained by pulling back the long thin cover from the coarse flowspace. To guide the reader through this proof we comment on thedependence of the appearing constants. In Theorem 3.1 we are given K ⊆ T compact and S ⊆ G finite. Together, K and S determine a HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 23 number ρ := d T ( K ∪ Sx , x ). Through the fellow traveler axiom (F2) ρ determines a number β in Lemma 3.10. Finally, the time τ for whichthe coarse flow is applied is provided in Lemma 3.11.3. a . Coarse flow spaces.
We set V := Gx ⊆ T . Note that since theaction of G on T is proper, V := V ∪ ∆ is a closed and therefore compactsubspace of T . We will define the coarse flow space as a subspace of V × V × ∆. Informally, it consists of all triples ( ξ − , v, ξ + ) for which v coarsely belongs to a quasi geodesic from ξ − to ξ + . Definition 3.2.
For K ⊆ T compact and ρ > CF ( K, ρ ) bethe subspace of V × V × ∆ consisting of all triples ( v − , v, ξ + ) ∈ V × V × ∆for which there is c ∈ G K with d T ( c (0) , v − ) ≤ ρ , d T (Im( c ) , v ) ≤ ρ and c ( ∞ ) = ξ + .We define the coarse ( K, ρ ) -flow space CF ( K, ρ ) as the closure of CF ( K, ρ ) inside of V × V × ∆. For ( x − , ξ + ) ∈ V × ∆ we define the coarse flow line between x − and ξ + as V K,ρ ( x − , ξ + ) := { v ∈ V | ( x − , v, ξ + ) ∈ CF ( K, ρ ) } ⊆ V. Lemma 3.3.
For K ⊆ T compact and ρ > there is R > with thefollowing property.(a) Let ( v − , ξ + ) ∈ V × ∆ with V K,ρ ( v − , ξ + ) = ∅ . Then there exists aquasi-geodesic ray c : [0 , ∞ ) → T such that the coarse flow line V K,ρ ( v − , ξ + ) is contained in the R -neighborhood of the image of c .(b) Let ( ξ − , ξ + ) ∈ ∆ × ∆ with V K,ρ ( ξ − , ξ + ) = ∅ . Then there ex-ists a quasi-geodesic c : R → T such that the coarse flow line V K,ρ ( ξ − , ξ + ) is contained in the R -neighborhood of the image of c .Here the additive constant for the quasi-geodesic (ray) c depends onlyon K and ρ , while the multiplicative constant is independent from K and ρ .Proof. (a) We use R from the fellow traveling axiom (F2). Still from (F2)we obtain for all t ∈ [0 , ∞ ) a neighborhood W t of ξ + in ∆ such that forall c, c ′ ∈ G K with d T ( c (0) , v − ) , d T ( c ′ (0) , v − ) ≤ ρ and c ( ∞ ) , c ′ ( ∞ ) ∈ W t we have d T ( c ( t ) , c ′ ( t )) ≤ R . As c is a quasi-geodesics with uniform con-stants, d T ( c ( n ) , c ( t )) is uniformly bounded for t ∈ [ n, n +1), and similarfor c ′ . After increasing the constant R , if necessary, we can assume W t is constant on intervals [ n, n + 1), n ∈ N . Now we can also assume,that for t ≥ t ′ we have W t ⊆ W t ′ . If V K,ρ ( v − , ξ + ) = ∅ , then for t ≥ c t ∈ G K with d T ( c t (0) , v − ) ≤ ρ and c t ( ∞ ) ∈ W t . We now definethe quasi-geodesic ray c by c ( t ) := c t ( t ). The multiplicative constant for c agrees with the multiplicative constant of the rays from G K , whilethe additive constant may increase by at most R . It is not difficult tocheck that V K,ρ ( v − , ξ + ) is contained in the R + ρ -neighborhood of theimage of c .(b) The quasi-geodesic c and R are provided by the infinite quasi-geodesic axiom (F3). Using in addition the small at ∞ axiom (F1)it follows that V K,ρ ( ξ − , ξ + ) is contained in the R -neighborhood of theimage of c . (cid:3) Lemma 3.4.
Let K ⊆ T compact and ρ > . Then for any ( x − , ξ + ) ∈ V × ∆ there exists a quasi isometric embedding V K,ρ ( x, ξ ) → Z .Moreover, the additive constant for this embedding depends only on K and ρ , while the multiplicative constant is also independent from K and ρ .Proof. This follows from Lemma 3.3 since the R -neighborhood of aquasi-geodesic ray is quasi-isometric to N ⊂ Z and the R -neighborhoodof a quasi-geodesic is quasi-isometric to Z . (cid:3) b . Long thin covers.
A subset W ⊆ V is said to be R -separatedif d T ( w, w ′ ) ≥ R for all w = w ′ ∈ W . A subset V ⊂ V is said to be( D, R )-doubling if the following holds for all R ≥ R : if W ⊆ V is R -separated and contained in a ball of radius 2 R , then the cardinalityof W is at most D . Proposition 3.5.
Let K ⊆ T be compact and ρ > .(a) dim CF ( K, ρ ) ≤ < ∞ .(b) For all ( x − , ξ + ) ∈ V × ∆ the set V K,ρ ( x − , ξ + ) is ( D, R ) -doubling.Here the constant D is independent of K , ρ , x − and ξ + , whilethe constant R depends on K and ρ , but not on x − and ξ − .(c) For each ( x − , v, ξ + ) ∈ CF ( K, ρ ) , the isotropy group G ( x − ,ξ + ) := { g ∈ G | g ( x − , ξ + ) = ( x − , ξ + ) } of ( x − , ξ + ) ∈ V × ∆ is virtuallycyclic.Proof. (a) As V and ∆ are separable and metrizable any subspace of V × V × ∆ is of dimension at most dim V + dim ∆ = 2 dim ∆.(b) The metric space Z is ( D, R )-doubling with D = 3, R = 0. It fol-lows that every space that quasi isometrically embeds into Z is ( D ′ , R ′ )-doubling with D ′ depending on D and the multiplicative constant ofthe quasi isometry and R ′ depending on D, R and the constants forthe quasi isometry. Therefore (b) is a consequence of Lemma 3.4.(c) The isotropy group G x − ,ξ + acts properly and isometrically on the HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 25 coarse flow line V L,ρ ( x − , ξ + ). Since V L,ρ ( x, ξ ) embeds quasi-isometri-cally into Z by Lemma 3.4 it follows that G x,ξ is virtually cyclic. (cid:3) We can now prove that our coarse flow spaces admit long thin covers.
Proposition 3.6.
There is a number N long such that for any K ⊆ T compact and any ρ, β > there exists a G -invariant cover U long ofCF ( K, ρ ) by open VCyc -sets such that the following two conditions aresatisfied:(a) the order of U long is at most N long ;(b) for any ( x − , v, ξ + ) ∈ CF ( K, ρ ) there is U ∈ U long with { x − }× B β ( v ) ×{ ξ + } ∩ CF ( K, ρ ) ⊆ U. Here B β ( v ) is the β -neighborhood of v in V .Proof. Using Proposition 3.5 this follows from [4, Thm. 1.1]. (cid:3) If U ∈ U long as above, then, typically, as a subset of V × V × ∆ the set U will be very small (i.e. thin) in the V - and the ∆-coordinate, whilethe coordinate in V varies over a subset that is long in the coarse flowlines (and thus long and coarsely thin).3. c . The coarse flow.
Informally, we have a coarse flow on the coarseflow space that moves towards ξ + along coarse flow lines. The partiallydefined coarse maps ι τ in the next definition should be thought of asthe composition of the coarse flow for time τ with the map ι thatsends ( g, ξ ) to the initial point gx in the coarse flow line V K,ρ ( gx , ξ ). Definition 3.7.
Let K ⊆ T compact and ρ ≥
0. For τ ≥ g, ξ ) ∈ G × ∆ we define ι τ ( g, ξ ) ⊆ V K,ρ ( gx , ξ )to consist of all v ∈ V for which there is c ∈ G K with d T ( c (0) , gx ) ≤ ρ , d T ( c ( τ ) , v ) ≤ ρ and c ( ∞ ) = ξ .For K ⊆ T compact and S ⊆ G finite we enlarge ( G × ∆) K to( G × ∆) SK := { ( gs, ξ ) | s ∈ S, ( g, ξ ) ∈ ( G × ∆) K } in order to have a space that contains gS ×{ ξ } for all ( g, ξ ) ∈ ( G × ∆) K .We now use ι τ to pull back open sets from the coarse flow space CF ( K, ρ ) to G × ∆ and to open subsets of ( G × ∆) SK . Definition 3.8.
Let K ⊆ T compact and ρ ≥ diam K ∪ { x } . For U ⊆ CF ( K, ρ ) and τ > ι − τ U ⊆ G × ∆ to consist of all ( g, ξ )for which { gx }× ι τ ( g, ξ ) ×{ ξ } ⊆ U For S ⊆ G finite we define ι − τS U ⊆ ( G × ∆) SK as the interior of ι − τ U ∩ ( G × ∆) SK in ( G × ∆) SK . If U is a collection ofopen subsets of CF ( K, ρ ), then we set ι − τS U := { ι − τS U | U ∈ U } .In the proof of Theorem 3.1 we will later use a collection of the form ι − τS U long , where U long comes from Proposition 3.6.The intersection with ( G × ∆) SK in the definition of ι − τS U is used toguarantee ι τ ( g, ξ ) = ∅ in the proof of the following lemma. Lemma 3.9.
Let K ⊆ T compact, S ⊆ G finite. Let ρ ≥ diam K ∪ Sx ∪ { x } . Let U, U ′ ⊆ CF ( K, ρ ) , τ > and γ ∈ G . Then(a) if U ∩ U ′ = ∅ , then ι − τS U ∩ ι − τS U ′ = ∅ ;(b) ι − τS ( γU ) = γ ( ι − τS U ) .Proof. We start with (a). Let ( g, ξ ) ∈ ι − τS U ∩ ι − τS U ′ . Then ( g, ξ ) ∈ ( G × ∆) SK and { gx }× ι τ ( g, ξ ) ×{ ξ } ⊆ U ∩ U ′ . It remains to show that ι τ ( g, ξ ) = ∅ . Since ( g, ξ ) ∈ ( G × ∆) SK there are s ∈ S and c ∈ G K with c (0) = gs − x , c ( ∞ ) = ξ . There is h ∈ G such that c ( τ ) ∈ hK ,since c ∈ G K . Now ρ ≥ diam K ∪ { x } implies d T ( c ( τ ) , hx ) ≤ ρ ,and ρ ≥ diamSx ∪ { x } implies d T ( c (0) , gx ) = d T ( gs − x , gx ) = d T ( x , sX ) ≤ ρ . Thus hx ∈ ι τ ( g, ξ ).Assertion (b) is a direct consequences of the definitions. (cid:3) d . Construction of U thick . By construction, the coarse flow linesare thickenings of the quasi geodesics from G K . In the next lemma wereplace one of the two quasi geodesics appearing in the fellow traveleraxiom (F2) with a coarse flow line. Lemma 3.10.
Let K ⊆ T be compact and ρ > . There is β > such that the following holds. Let c ∈ G K and τ ≥ and such that c ( τ ) ∈ K . Then there exists a neighborhood W of ξ := c ( ∞ ) such thatfor all ξ ′ ∈ W , g ′ ∈ G , v ′ ∈ ι τ ( g ′ , ξ ′ ) with d T ( c (0) , g ′ x ) ≤ ρ we have d T ( x , v ′ ) ≤ β. Proof.
We apply the fellow traveler axiom (F2) for 2 ρ and obtain anumber R ≥
0. Still from (F2) we obtain for x := c (0), ξ := c ( ∞ ), and t := τ a neighborhood W of ξ . Let now ξ ′ ∈ W , g ′ ∈ G , v ′ ∈ ι τ ( g ′ x , ξ ′ )with d T ( c (0) , g ′ x ) ≤ ρ . Then there is c ′ ∈ G K with d T ( c ′ (0) , g ′ x ) ≤ ρ , d T ( c ′ ( τ ) , v ′ ) ≤ ρ and c ′ ( ∞ ) = ξ ′ . Now d T ( c (0) , c ′ (0)) ≤ d T ( c (0) , g ′ x ) + d T ( g ′ x , c ′ (0)) ≤ ρ . Therefore, the assertion of the fellow traveling HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 27 property yields d T ( c ( τ ) , c ′ ( τ )) ≤ R . This implies d T ( x , v ′ ) ≤ d T ( x , c ( τ )) + d T ( c ( τ ) , c ′ ( τ )) + d T ( c ′ ( τ ) , v ′ ) ≤ diam K ∪ { x } + R + ρ =: β. (cid:3) To prove Theorem 3.1 we will use a β -long thin cover U long fromProposition 3.6 where β comes from Lemma 3.10. In the next lemma weuse the small at ∞ axiom (F1) to show that for sufficiently large τ thepull back of U long with ι τ to ( G × ∆) SK is S -long for all ( g, ξ ) ∈ ( G × ∆) K ,i.e., it satisfies the assertion (b) in Theorem 3.1. Lemma 3.11.
Let K ⊆ T be compact and S ⊆ G be finite. Let ρ ≥ diam Sx ∪ { x } ∪ K . Let β be as in Lemma 3.10.Let U long be the cover of CF ( K, ρ ) appearing in Proposition 3.6. Thenthere is τ > such that for all ( g, ξ ) ∈ ( G × ∆) K there is U ∈ U long with gS ×{ ξ } ⊆ ι − τS U. Proof.
We argue by contradiction and assume that the assertion fails.Then, for τ → ∞ , there are ( g τ , ξ τ ) ∈ ( G × ∆) K such that(3.12) g τ S ×{ ξ τ } 6⊆ ι − τS U for all U ∈ U long . Since ( g τ , ξ τ ) ∈ ( G × ∆) K there is c τ ∈ G K with c τ (0) = g τ x and c τ ( ∞ ) = ξ τ . Since U long is G -invariant, we may assume that c τ ( τ ) ∈ K for all τ . Using ρ ≥ diam K ∪ { x } , we obtain x ∈ V K,ρ ( g τ x , ξ τ ) forall τ .Since V and ∆ are compact we can, after a subsequence, assumethat lim τ →∞ ( g τ x , x , ξ τ ) = ( ξ − , x , ξ + ) ∈ CF ( K, ρ )exists. By Proposition 3.6 there is U ∈ U long such that { ξ − }× B β ( x ) ×{ ξ + } ∩ CF ( K, ρ ) ⊆ U. As U is open and B β ( x ) is finite there are open neighborhoods U − ⊆ V of ξ − and U + ⊆ ∆ of ξ + such that(3.13) U − × B β ( x ) × U + ∩ CF ( K, ρ ) ⊆ U. The small at ∞ axiom (F1) implies that for all s ∈ S eventually g τ sx ∈ U − . We find now τ such that g τ sx ∈ U − for all s ∈ S and ξ τ ∈ U + .We claim that(3.14) g τ S ×{ ξ τ } ⊆ ι − τS U. This will contradict (3.12). To prove (3.14) we apply Lemma 3.10 to c := c τ and obtain a neighborhood W of ξ τ in ∆. After shrinking W we may assume W ⊆ U + . We claim that(3.15) g τ S × W ∩ ( G × ∆) SK ⊆ ι − τ U which will imply (3.14). Let s ∈ S and ξ ′ ∈ W with ( g τ s, ξ ′ ) ∈ ( G × ∆) SK . We need to show that { g τ sx }× ι τ ( g τ s, ξ ′ ) ×{ ξ ′ } ⊆ U. Let v ′ ∈ ι τ ( g τ s, ξ ′ ). Since ξ ′ ∈ W and since d T ( c τ (0) , g τ sx ) = d T ( g τ x , g τ sx ) ≤ ρ we can use the assertion of Lemma 3.10 (for g ′ := g τ s ) to obtain d T ( x , v ′ ) ≤ β . Since g τ sx ∈ U − and since ξ ′ ∈ W ⊆ U + we ob-tain ( g τ sx , v ′ , ξ ′ ) ∈ U from (3.13). Thus (3.15) holds. (cid:3) Proof of Theorem 3.1.
Set N thick := N long , where N long is from Propo-sition 3.6. Let S ⊆ G finite and K ⊆ T compact be given. Let ρ := diamK ∪ Sx ∪ { x } . Let β as in Lemma 3.10. Let U long bethe cover of CF ( K, ρ ) appearing in Proposition 3.6. As U consists ofVCyc-subsets and is G -invariant, the same holds for ι − τS U long for any τ by Lemma 3.9. Lemma 3.9 also implies that for any τ , the orderof ι − τS U long , does not exceed the order of U long . By definition ι − τS con-sists of open subsets of ( G × ∆) SK . As ∆ is metrizable there exists a G -invariant metric on G × ∆. This allows us to extend each V ∈ ι − τS U to an open subset V ′ ⊆ G × ∆ such that U thick := { V ′ | V ∈ ι − τS U } also consists of VCyc-subsets, also is G -invariant and also is of orderat most N thick , see Lemma 3.16 below for more details. Finally, thereis, by Lemma 3.11, τ > g, ξ ) ∈ ( G × ∆) K there is V = ι − τS U with gS ×{ ξ } ⊆ V ⊆ V ′ ∈ U thick . (cid:3) Lemma 3.16.
Let X be a G -space and Y be a G -invariant subspace.Assume that the topology on X can be generated by a G -invariant met-ric d . Let U be a G -invariant collection of open F -subsets of Y . Thenthere exits a G -invariant collection U + of open F -subsets of X suchthat(a) the order of U + equals the order of U ;(b) for each U ∈ U there is U + ∈ U + with U = U + ∩ Y .Proof. For U ∈ U set U + := { x ∈ X | d ( x, U ) < d ( x, Y r U ) } . It is notdifficult to check that U + := { U + | U ∈ U } has the required properties,see for example [4, App. B]. (cid:3) HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 29 Finitely F -amenable actions. a . Finite extensions and N - F -amenability. The main results inthis section are Propositions 4.4 and 4.5. While in principle it is pos-sible to prove these directly from the definition, we find it convenientto reformulate N - F -amenability in terms of maps to simplicial com-plexes and prove the statements from this point of view. The coversone would naturally write down from the definition would not be F -covers, corresponding to actions on simplicial complexes with elementsthat leave a simplex invariant without fixing it pointwise. This is fixedby barycentrically subdividing, while the operation on covers is lesstransparent.Let E be a simplicial complex with vertex set V ( E ). Every point of E can be written as y = P v ∈ V C y v · v with y v ∈ [0 ,
1] and P v ∈ V ( E ) y v = 1.The ℓ -metric on E is defined by d E ( y, y ′ ) := P v ∈ V ( E ) | y v − y ′ v | . Webriefly discuss products of simplicial complexes. Let E be a simpli-cial complex. For n ∈ N we define a simplicial structure on the N -fold cartesian product E × n of E as follows: First we replace E by itsbarycentric subdivision. This is a locally ordered simplicial complex;for each simplex the set of its vertices has a linear order and this orderis compatible with taking faces of simplices. The product of locallyordered simplicial complexes is canonically a simplicial complex. Fora simplicial action of a group G on E the product action on E × n isalso simplicial. On E × n it will be convenient to use the product metric d E × n defined by d E × n (( y , . . . , y n ) , ( y ′ , . . . , y ′ n )) = max ≤ i ≤ n d E ( y i , y ′ i ).This is not the ℓ -metric d E × n , but the change is uniformly controlled,provided that E is finite dimensional. Lemma 4.1.
Fix n, N ∈ N . Then for any ε > there is ε > suchthat for any simplicial complex E of dimension at most N we have forall y = ( y , . . . , y n ) , y ′ = ( y ′ , . . . , y ′ n ) ∈ E × n d E × n ( y, y ′ ) < ε = ⇒ d E × n ( y, y ′ ) < ε,d E × n ( y, y ′ ) < ε = ⇒ d E × n ( y, y ′ ) < ε. Proof. If E = ∆ N ′ , then this is a consequence of compactness of(∆ N ′ ) × n . But this case implies the general case for the followingreason. Let y, y ′ ∈ E × n be given. Then there are simplices σ i , σ ′ i , i = 1 , . . . , n of E with y i ∈ σ i , y ′ i ∈ σ ′ i . Let F ⊂ E be the subcom-plex spanned by the σ i . This subcomplex has at most 2 n ( N + 1)-manyvertices and embeds therefore into an N ′ -simplex σ ∼ = ∆ N ′ , where N ′ = 2 n ( N + 1) −
1. Since d E × n ( y, y ′ ) = d F × n ( y, y ′ ) = d σ × n ( y, y ′ ) and d E × n ( y, y ′ ) = d F × n ( y, y ′ ) = d σ × n ( y, y ′ ), the general case follows. (cid:3) Let E be a simplicial complex equipped with a simplicial G -actionand ∆ be a G -space. For S ⊂ G finite and ε > f : ∆ → E is said to be ( S, ε ) -equivariant ifsup x ∈ ∆ ,s ∈ S d E ( sf ( x ) , f ( sx )) < ε. Let F be a family of subgroups of G . By an ( G, F )-simplicial complexwe mean a simplicial complex E with a simplicial G -action such thatall isotropy groups belong to F . Lemma 4.2.
Let F be a family of subgroups of G and ∆ be a compactmetrizable space with a G -action. Then the following are equivalent.(a) The action of G on ∆ is N - F -amenable;(b) For any S ⊆ G finite and ε > there exists a ( G, F ) -simplicialcomplex of dimension at most N and an ( S, ε ) -equivariant map ∆ → E .Proof. This is proven in [35, Prop. 4.2] with the following two minorchanges: Firstly, in [35] the covers of G × ∆ are in addition assumed tobe cofinite for the action of G . However, since ∆ is compact, it is alwayspossible to pass to a cofinite subcover. Secondly, in [35] the family F isassumed to be closed under taking supergroups of finite index. This hasthe advantage that the isotropy groups for a simplicial action belongto F if and only if the isotropy groups of all vertices belong to F . Forgeneral F this is only true for cellular actions. However, the inducedaction on the nerve of an F -cover is cellular. Therefore [35, Prop. 4.2]remains true without the assumption that F is closed under takingsupergroups of finite index. (cid:3) We will use Lemma 4.2 to discuss the behavior of N - F -amenabilityunder finite extensions. This is closely related to [11, Sec. 5]. As apreparation we discuss coinduction. Let G ⊂ G be a subgroup offinite index. For a G -space E we set E := map G ( G, E ); this isthe coinduction of E from G to G . We obtain an action of G on E as follows: the action of g ∈ G on ( y : G → E ) ∈ E is given bythe formula ( gy )( a ) := y ( ag ) for all a ∈ G . If G y is the isotropy sub-group of G for ( y : G → E ) ∈ E , then G y ∩ G is contained in theisotropy subgroup ( G ) y ( e ) of G for y ( e ) ∈ E . If E is a simplicialcomplex, then E is a simplicial complex with the following construc-tion. First we replace E by its barycentric subdivision. Now it isa locally ordered simplicial complex and the action of G preservesthis order. Write V ( E ) for the vertices of E and define the ver-tices of E by V ( E ) := map G ( G, V ( E )). Now v , . . . , v n ∈ V ( E )span a simplex for E if for each g ∈ G the vertices v ( g ) , . . . , v n ( g ) HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 31 span a simplex (possibly of dimension < n ) of E and in the lo-cal order of this simplex we have v ( g ) ≤ v ( g ) ≤ · · · ≤ v n ( g ). Itwill be convenient to use the G -invariant metric d E on E defined by d E ( y, y ′ ) := max a ∈ G d E ( y ( a ) , y ′ ( a )). This is not the ℓ -metric on E .However, since, forgetting the G -action, E = E × m for m = [ G : G ],Lemma 4.1 implies that if E is finite dimensional, then the identityon E is in both directions between ( E, d E ) and ( E, d E ) uniformly con-tinuous.If f : ∆ → E is G -equivariant, then we obtain a G -mapmap G ( G, ∆ ) → map G ( G, E ) , ξ f ◦ ξ. But if f is not G -equivariant (maybe only ( S, ε )-equivariant), thenthis only defines a map map G ( G, ∆ ) → map( G, E ) . If G = G t ⊔· · · ⊔ G t n , then there is a projection π : map( G, E ) → map G ( G, E )determined by π ( y )( t i ) = y ( t i ) for i = 1 , . . . , n . We write now f : map G ( G, ∆ ) → map G ( G, E )for the map π ◦ ( f ) ∗ ; it is determined by ( f ( ξ ))( t i ) = f ( ξ ( t i )) for i = 1 , . . . , n . Lemma 4.3.
For any finite S ⊂ G and ε > there are S ⊂ G finiteand ε > such that the following holds. Suppose that f : ∆ → E is ( S , ε ) -equivariant where dim E ≤ N , then f : map G ( G, ∆ ) → map G ( G, E ) =: E with ( f ( ξ ))( t i ) = f ( ξ ( t i )) for i = 1 , . . . , n is ( S, ε ) -equivariant.Proof. Because of Lemma 4.1 we can use the metric d E instead of d E .Note that d E ( y, y ′ ) = max i d E ( y ( t i ) , y ′ ( t i )), since the action of G on E is isometric for d E .Let S ⊂ G be finite. Set S := { t i st − j | s ∈ S, ≤ i, j ≤ n } ∩ G .Let f : ∆ → E be ( S , ε )-equivariant. For s ∈ S , ξ ∈ map G ( G, ∆ )and t i we pick t j with t i s ∈ G t j , thus t i st − j ∈ G . Then( f ( sξ ))( t i ) = f (( sξ )( t i )) = f ( ξ ( t i s ))= f ( ξ ( t i st − j t j )) = f ( t i st − j ξ ( t j ))( s ( f ( ξ )))( t i ) = ( f ( ξ ))( t i s ) = ( f ( ξ ))( t i st − j t j )= t i st − j ( f ( ξ )( t j )) = t i st − j f ( ξ ( t j ))and therefore d E (( f ( sξ ))( t i ) , ( s ( f ( ξ )))( t i )) < ε . It follows that d E ( f ( sξ ) , s ( f ( ξ ))) < ε and that f is ( S, ε )-equivariant. (cid:3)
Proposition 4.4.
Let G act on the compact metrizable space ∆ . Let G be a subgroup of finite index n in G and let F be a family ofsubgroups of G . Suppose that the restriction of the action on ∆ to thesubgroup G is N - F -amenable. Then the action of G on ∆ is n · N - F -amenable, where F is the family of subgroups F of G for which F ∩ G belongs to F .Proof. It suffices to show that condition (b) from Lemma 4.2 passesfrom G to G . Let S ⊆ G be finite and ε > for ∆ with the action restricted to G . Let f : ∆ → E be ( S , ε )-equivariant where E is an ( G , F )-simplicial complex of dimensionat most N and S ⊆ G finite and ε > E := map G ( G, E ); this is an ( G, F )-simplicial complex of dimen-sion ≤ n · N . By Lemma 4.3 there exists an ( S, ε )-equivariant map f : map G ( G, ∆ ) → E . Composing f with the G -equivariant map∆ → map G ( G, ∆ ), ξ ( g gξ ) we obtain a ( S, ε )-equivariant map∆ → E . (cid:3) Proposition 4.5.
Let G be a subgroup of G of finite index n . Let F be a family of subgroups of G . Let F be the family of subgroups F of G for which F ∩ G belongs to F . Assume that there exists an N - F -amenable action of G on a compact metrizable space ∆ . Thenthe induced action of G on map G ( G, ∆ ) ∼ = ∆ n is n · N - F -amenable.Proof. If E is an ( G , F )-simplicial complex of dimension N , then E := map G ( G, E ) is an ( G, F )-simplicial complex of dimension n · N .The result follows from Lemma 4.3 and Lemma 4.2. (cid:3) b . The Farrell-Jones Conjecture.
Let G be a group. Let A be anadditive category with a strict G -action and a strict direct sum. Fol-lowing [7, Sec. 4.1] we will call such a category an additive G -category.For such a category A there is then an additive category R G A . If A isequivalent to the category of finitely generated free R -modules and the G -action is trivial, then R G A is equivalent to the category of finitelygenerated free R [ G ]-modules. Given a family F of subgroups of G thereis the K -theoretic F -assembly map α KG, F : H G ∗ ( E F G ; K A ) → K ∗ ( R G A ) . The K -theoretic Farrell-Jones Conjecture (with coefficients) assertsthat this map is an isomorphism if we use for F the family VCycof virtually cyclic subgroups [12, Conj. 3.2]. Farrell and Jones’ originalformulation [30] for the group ring Z [ G ] is a special case of this moregeneral formulation. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 33 If A is in addition equipped with a strict involution [7, Sec. 4.1], then R G A inherits an involution and there is for any family F of subgroupsof G the L -theoretic F -assembly map α LG, F : H G ∗ ( E F G ; L −∞A ) → L h−∞i∗ ( R G A ) . The L -theoretic Farrell-Jones Conjecture (with coefficients) asserts thatthis map is an isomorphism if we use for F the family VCyc of virtuallycyclic subgroups [6]. Again, Farrell and Jones’ original formulation [30]for the group ring Z [ G ] is a special case of this more general formula-tion.We will say that a group G satisfies the Farrell-Jones Conjecturerelative to F if for all G -additive categories A (with strict involution inthe L -theory case) the assembly maps α KG, F and α LG, F are isomorphisms.If F = VCyc then we will say that G satisfies the Farrell-Jones Con-jecture . We will need the following results. Theorem 4.6 (Transitivity principle) . Let F be a family of subgroupsof G . Assume that G satisfies the Farrell-Jones Conjecture relative to F and that any subgroup F ∈ F satisfies the Farrell-Jones Conjecture.Then G satisfies the Farrell-Jones Conjecture.Proof. See for example [5, Thm. 2.10]. (cid:3)
Remark . An application of the Transitivity principle 4.6 is the fol-lowing inheritance property for the Farrell-Jones conjecture for exten-sions, see for example [5, Thm. 2.7]. Let N → ˆ G → G be an exten-sion. Suppose that G satisfies the Farrell-Jones Conjecture and thatthe preimage in ˆ G of any virtually cyclic subgroup of G also satisfiesthe Farrell-Jones Conjecture. Then ˆ G satisfies the Farrell-Jones Con-jecture.In the next result ER stands for Euclidean retract. Recall that acompact space X is a Euclidean retract (or ER) if it can be embeddedin some R n as a retract. A compact metrizable space X is an ER ifand only if it is a finite-dimensional contractible ANR. Theorem 4.8.
Let F be a family of subgroups of G that is closedunder taking finite index overgroups. Suppose that G admits a finitely F -amenable action on a compact ER. Then G satisfies the Farrell-Jones Conjecture relative to F .Proof. This follows from the main axiomatic results of [7, 9]. Theassumptions on G are formulated somewhat differently in these refer-ences, but it is not hard to check that the present assumptions implythe assumptions in these references, see also [4, Thm. 4.3]. (cid:3) If F is a class of groups that is closed under taking subgroups andisomorphism, then for a group G we denote by F ( G ) the family ofsubgroups of G that belong to F . For such a class of groups F we definethe class ac ( F ) of groups to consist of all groups G that admit a finitely F ( G )-amenable action on a compact ER . Using the action on a pointwe see F ⊆ ac ( F ). Lemma 4.9.
Let F be a class of groups that is closed under isomor-phisms, taking subgroups, taking finite index overgroups, finite productsand central extensions with finitely generated kernel. Then(a) if all groups in F satisfy the Farrell-Jones Conjecture, then allgroups in ac ( F ) satisfy the Farrell-Jones Conjecture;(b) the class ac ( F ) is again closed under isomorphisms, taking sub-groups, taking finite index overgroups, finite products and centralextensions with finitely generated kernel.Proof. (a) By Theorem 4.8 every group G from ac ( F ) satisfies theFarrell-Jones Conjecture relative to F ( G ). By assumption every groupfrom F ( G ) satisfies the Farrell-Jones Conjecture. Therefore the transi-tivity principle 4.6 implies that G satisfies the Farrell-Jones Conjecture.(b) That ac ( F ) is closed under isomorphism and taking subgroups isclear from the definition. Proposition 4.5 implies that it is also closedunder finite index overgroups.Let for i = 1 , G i act finitely F i -amenable on ∆ i . Thenthe product action of G × G on ∆ × ∆ is finitely F ×F -amenable.Since F is assumed to be closed under finite products it follows that ac ( F ) is also closed under finite products.Let C → ˆ G → G be a central extension with C finitely generated. If G acts finitely F -amenable on ∆, then ˆ G acts via the projection ˆ G → G finitely F ′ -amenable on ∆, where F ′ consists of central extensions withfinitely generated kernel of groups in F . Therefore, if F ⊆ F , then also F ′ ⊆ F . It follows that ac ( F ) is closed under central extensions withfinitely generated kernel. (cid:3) Starting with ac ( F ) := F we can define inductively ac n +1 ( F ) := ac ( ac n ( F )). We set AC ( F ) := S ac n ( F ). Corollary 4.10.
Let F be a class of groups that is closed under iso-morphisms, taking subgroups, taking finite index overgroups and finiteproducts. Assume that all groups from F satisfy the Farrell-Jones Con-jecture. Then all groups from AC ( F ) satisfy the Farrell-Jones Conjec-ture.Proof. This follows by induction from Lemma 4.9. (cid:3)
HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 35
Let
VNil be the class of finitely generated virtually nilpotent groups.Later, in Lemma 9.3, we will use Theorem B to show that mapping classgroups of surfaces belong to AC ( VNil ). Thus, to prove Theorem A wewill need the following well-known result.
Proposition 4.11.
All groups in the class
VNil satisfy the Farrell-Jones Conjecture.Proof.
Finitely generated virtually abelian groups satisfy the Farrell-Jones Conjecture, see for example [5, Thm. 3.1].Let N → ˆ G → G be an extension. If N is finitely generated and cen-tral, then all preimages of virtually cyclic groups are virtually finitelygenerated abelian. The inheritance property from Remark 4.7 nowimplies that the Farrell-Jones Conjecture is stable under central exten-sions with finitely generated kernel.The case of finitely generated virtually nilpotent groups follows nowby induction on the length of the lower central series. This inductionis carried out in detail in an only marginally different situation in [10,Lem. 2.13]. (cid:3) Remark . All virtually nilpotent subgroups of the mapping classgroup are known to be virtually abelian, see [19] and [43, Theorem8.9]. Thus it may seem weird that nilpotent groups come up in ourargument. Indeed, after a reorganization of the induction process wecould avoid mentioning nilpotent groups, but we would still need thefact that central extensions (with finitely generated free abelian kernel)of groups satisfying the Farrell-Jones Conjecture satisfy the Farrell-Jones Conjecture. As explained above, the Farrell-Jones Conjecturefor virtually nilpotent groups is an easy consequence of this fact andof the Farrell-Jones Conjecture for virtually abelian groups.
Remark . Lemma 4.9 (and its proof) remains true if we replace cen-tral extension with finitely generated kernel with extension with abeliankernel . Since Wegner [74] proved the Farrell-Jones Conjecture for theclass
VSol of virtually solvable groups it follows that all groups in AC ( VSol ) satisfy the Farrell-Jones Conjecture as well. Wegner’s proofis considerably more involved than the proof of the Farrell-Jones Con-jecture for virtually finitely generated nilpotent groups given above.It is not clear that AC ( VSol ) or AC ( VNil ) have all the inheritanceproperties known for the Farrell-Jones Conjecture. For example theFarrell-Jones Conjecture is also known to be stable under directed col-imits and finite free products. If a group G acts finitely F -amenablyon a compact ER , then G is (strongly) transfer reducible relative to F in the sense of [7, 73]. Groups that are transfer reducible relative to groups that satisfy the Farrell-Jones Conjecture satisfy the FarrellJones Conjecture themselves. This is a consequence of the transitivityprinciple 4.6 and the main axiomatic results from [7, 73].5. Preliminaries on mapping class groups
The general references for this section are [28, 31, 41]. Let Σ bea closed oriented surface of genus g , with p ≥ P ⊂ Σ the set of punctures. The mapping class group
Mod(Σ) = π ( Homeo + (Σ , P ))is the group of components of the group of orientation-preserving home-omorphisms of Σ that leave P invariant.We will always assume 6 g + 2 p − > r P .The sporadic cases are g = 0, p ≤ g = 1, p = 0 when Mod(Σ) = SL ( Z ) is virtually free.A simple closed curve in Σ r P is essential if it does not bound a diskor a once punctured disk. We denote by S the set of isotopy classesof essential simple closed curves in Σ r P and refer to its elementsas curves . If Σ r P is given a complete hyperbolic structure of finitearea, every s ∈ S has a unique geodesic representative. If s, s ′ ∈S the intersection number i ( s, s ′ ) is the smallest cardinality of a ∩ a ′ as a, a ′ range over simple closed curves in the isotopy classes s, s ′ respectively. Thus i ( s, s ) = 0 and for s = s ′ i ( s, s ′ ) is the cardinality ofthe intersection between the geodesic representatives of s, s ′ . See [28,Section 1.2] or [24, Lemma 2.6].To Σ one associates several spaces on which the mapping class groupMod(Σ) acts.5. a . Teichm¨uller space.
The
Teichm¨uller space T = T (Σ) is thespace of marked complex structures on Σ with P the set of distin-guished points. Equivalently, by the Uniformization Theorem, T isthe space of marked complete hyperbolic structures of finite area onΣ r P . Then T is naturally a smooth (or even complex analytic) mani-fold diffeomorphic to R g +2 p − . The mapping class group Mod(Σ) actsby changing the marking. This action is discrete but not cocompact;however, there are natural cocompact subspaces. Fix an ǫ > thick part T ≥ ǫ ⊂ T consisting of X ∈ T such that everyclosed hyperbolic geodesic has length ≥ ǫ . Theorem 5.1 (Mumford [64]) . The thick part T ≥ ǫ ⊂ T is cocompact. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 37
Thus for ǫ n ց
0, the sequence T ≥ ǫ n forms an exhaustion of T bycocompact subsets.For X ∈ T we use the marking to identify the set of curves on X with S .5. b . Measured foliations.
This is an important tool introduced byThurston, see [31]. It provides a “completion” of the set S , much likethe circle is a completion of Q ∪ {∞} .A measured foliation on Σ is a foliation with finitely many singulari-ties equipped with a transverse measure of full support. The singulari-ties are standard k -prong singularities with k ≥ k = 1 isallowed at the punctures. Each leaf is either an arc joining two singu-larities or punctures (that may coincide), or an essential circle, or aninjectively immersed line or ray that starts at a puncture or a singularpoint.The Whitehead equivalence on the set of measured foliations is gen-erated by collapsing leaves that are arcs joining distinct singularitiesand isotopies. Every measured foliation µ determines a length func-tion ℓ µ : S → [0 , ∞ ) that sends s ∈ S to the infimum of measures oversimple closed curves in the isotopy class s .The set of all measured foliations up to Whitehead equivalence has anatural topology, homeomorphic to R g +2 p − −{ } . It is defined by em-bedding the set of measured foliations in the space of length functions ℓ : S → [0 , ∞ ). Adding the “empty foliation” 0, one obtains the space MF homeomorphic to R g +2 p − , and projectivizing with respect to theaction of R + that scales the measure, the space PMF homeomorphicto S g +2 p − .Every curve s ∈ S also determines a length function ℓ s : S → [0 , ∞ )via ℓ s ( s ′ ) = i ( s, s ′ ). There is a unique equivalence class j ( s ) of measuredfoliations such that ℓ j ( s ) = ℓ s . The function j : S ֒ → MF is the canonical inclusion . The foliation j ( s ) has all but finitely many leavesin the isotopy class s . We will usually suppress j and write S ⊂ MF .The subset
S ⊂ MF is closed and discrete, but after projectivizing,the image of S in PMF is dense. The intersection pairing on S extendsuniquely to i : MF × MF → [0 , ∞ ) in such a way that it is continuousand R + -equivariant in each variable. Moreover, i is symmetric and i ( µ, s ) = ℓ µ ( s ) when s ∈ S , and i ( µ, µ ) = 0 for every µ ∈ MF . Theintersection pairing does not descend to PMF ; however the statement i ( ξ, η ) = 0 (or = 0) makes sense for projectivized measured foliations ξ, η .A measured foliation µ is filling if i ( µ, s ) > s ∈ S . This isequivalent to the condition that no curve s ∈ S can be homotoped into the union of finitely many leaves. If µ is filling, the set ∆( µ ) = { [ ν ] ∈PMF | i ( µ, ν ) = 0 } has the structure of a simplex [48, Theorem 14.7.6]and consists of classes of measures with the same underlying foliationas µ (for the latter see [70, Theorem 1.12]). The vertices correspond toergodic measures, and general points to convex combinations of ergodicmeasures. One characterization of an ergodic measure µ is that if it iswritten as the sum µ = µ + µ of transverse measures then necessarilyboth µ i are multiples of µ . When the simplex degenerates to a point, µ is called uniquely ergodic .Thurston [31] constructed an equivariant compactification T of T such that T − T = PMF and the pair ( T , T ) is homeomorphic to thepair ( B, intB ) where B is the closed ball of dimension 6 g + 2 p − X ∈ T ,thought of as a hyperbolic surface, determines a length function ℓ X bysending s ∈ S to its hyperbolic length. Then a sequence of hyperbolicsurfaces converges to the projective class of a measured foliation ifthe corresponding length functions converge projectively to the lengthfunction of the foliation.5. c . Measured geodesic laminations. A geodesic lamination in acomplete hyperbolic surface Σ of finite area is a nonempty compactsubset of Σ r P which is a disjoint union of geodesics (as a set). A measured geodesic lamination ξ is a geodesic lamination equipped witha transverse measure. To an arc A transverse to the lamination andwith endpoints in the complement, ξ assigns a real number R A ξ ≥ s ∈ S we have the intersection number i ( s, ξ ) defined as the in-fimum of R A ξ as A ranges over (transverse) parametrizations of simpleclosed curves in the isotopy class s . There is a natural bijection be-tween the set MF of measured foliations up to isotopy and Whiteheadmoves, and the set ML of measured geodesic laminations. See [53] and[46, Chapter 11]. If a measured foliation µ corresponds to the measuredlamination ξ then i ( s, µ ) = i ( s, ξ ) for every s ∈ S . A rough descriptionof the correspondence is as follows. Let ℓ be a generic leaf of µ . Its liftto the universal cover of Σ r P (which can be identified with hyper-bolic plane via a complete finite area hyperbolic metric on Σ r P ) is aquasi-geodesic which is bounded distance away from a unique infinitegeodesic l . The image of l is a generic leaf of ξ .When ξ is a measured geodesic lamination, denote by | ξ | its support,i.e. the union of those leaves of ξ such that the measure of any arccrossing it transversally is nonzero. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 39 d . The supporting multisurface.
Consider a measured geodesiclamination ξ . The support | ξ | is a geodesic lamination with finitelymany components and each is minimal (i.e. every leaf is dense), in-cluding the possibility of a simple closed geodesic. Since we requirethat | ξ | be compact, there are no leaves going to punctures. Even moregenerally, a geodesic lamination (possibly not the support of a mea-sure) consists of finitely many minimal components and finitely manyisolated leaves, each of which is either closed or in each direction spiralstowards a closed leaf or a minimal component. See e.g. [20, Propo-sition 3]. The spiraling leaves cannot be in the support of a measuresince they would give rise to transverse arcs with infinite measure.We say that ξ or | ξ | is filling if every complementary component of | ξ | is homeomorphic to an open disk or to an open once punctureddisk. Equivalently, every simple closed geodesic α in Σ intersects | ξ | ,or equivalently again, i ( α, ξ ) >
0, i.e. the corresponding measuredfoliation is filling. If ξ is filling then | ξ | is connected.Unless otherwise stated, when we talk about subsurfaces Y ⊂ Σ wemean • connected and closed, as subsets of Σ, • no punctures on the boundary, • Y = Σ, • Y is not a disk or a once punctured disk or a pair of pants, bywhich we mean a sphere with the total of exactly three puncturesand boundary components, • no complementary component is a disk or a punctured disk, • up to isotopy rel P .In particular, subsurfaces are π -injective.A multisurface is a nonempty disjoint union of subsurfaces that doesnot contain distinct annuli which are isotopic rel P .When | ξ | is connected but not filling there is a unique subsurface Y ⊂ Σ that contains | ξ | and ξ is filling in Y . We call Y the sup-porting subsurface of ξ and denote it Supp ( ξ ) or Supp ( | ξ | ). That ξ fills Supp ( | ξ | ) means that i ( s, ξ ) > s in Supp ( | ξ | )not homotopic into ∂ ( Supp ( | ξ | )).We note that the supporting subsurface of a simple closed curve is anannulus, and otherwise the supporting subsurface has negative Eulercharacteristic and cannot be a pair of pants (from the point of view offoliations this was proved in [31, Expos´e 6]).In general, when | ξ | is disconnected, the supporting subsurfaces ofthe components can be isotoped so that they are pairwise disjoint. The union of these supporting subsurfaces of the components is by defini-tion the supporting multisurface Supp ( ξ ) or Supp ( | ξ | ). (The annulicomponents correspond to closed geodesics in | ξ | and are pairwise notisotopic rel P .)Now that we made the careful distinction, we will revert to thestandard terminology and call Supp ( ξ ) the supporting subsurface evenwhen it is not connected.The set of geodesic laminations in Σ is a compact space with respectto Hausdorff topology on the space of compact subsets of Σ. Thefollowing is standard. Proposition 5.2.
Suppose ξ n → ξ is a convergent sequence of mea-sured geodesic laminations, and suppose that | ξ n | → λ in the Hausdorfftopology. Then | ξ | ⊆ λ .Proof. Let s be a curve in the complement of Supp ( λ ). Then i ( s, ξ n ) =0 for large n since s is disjoint from Supp ( ξ n ). It follows that i ( s, ξ ) = 0,so s is disjoint from the support of ξ . (cid:3) e . The Teichm¨uller metric.
The Teichm¨uller space T is equippedwith a proper geodesic metric which is Mod(Σ)-invariant. The distancebetween two complex surfaces is defined to be d T ( X, Y ) = inf log( K f )where f ranges over all orientation preserving homeomorphisms X → Y which are smooth except at finitely many points, and K f = sup K f ( p )is the supremum of dilatations K f ( p ) over the points p ∈ X where f issmooth (it is customary to scale this expression by but we will ignorethis). Recall that K f ( p ) ≥ df p . Teichm¨uller proved that the infimum of K f is realized by a uniquehomeomorphism, called the Teichm¨uller map . The Teichm¨uller metricis proper and Mod(Σ)-invariant.5. f . Holomorphic quadratic differentials.
The cotangent space of T at X ∈ T is the space of (holomorphic) quadratic differentials on X ,each of which is defined in charts as q ( z ) = f ( z ) dz with f holomorphic,possibly with simple poles at the punctures. See e.g. [42] for basic factsabout quadratic differentials.A nonzero quadratic differential q on X ∈ T determines two mea-sured foliations, horizontal q H and vertical q V . Away from the singu-larities, i.e. points where q has a zero or a pole, there are charts where HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 41 q = dz , and then the vertical foliation is defined by the vertical linesand with transverse measure | dx | , and similarly for the horizontal folia-tion. When s is a curve, we will say its horizontal length is i ( s, q V ), themeasure assigned to s by the vertical foliation, and we similarly definethe vertical length of s . The quadratic differential q also determines aEuclidean metric on Σ with cone type singularities: on a chart where q = dz the metric is Euclidean. We will denote this metric by l q .The norm of q is || q || = R X | q | , i.e. it equals the area of X withrespect to the Euclidean metric. We denote by QD ( X ) the vectorspace of all quadratic differentials on X and by QD ( X ) the subsetof unit norm quadratic differentials. The following fact can be provedusing the compactness of the unit area quadratic differentials on X . Lemma 5.3.
Let X be a hyperbolic surface. There is ǫ X > suchthat for every q ∈ QD ( X ) and every curve s we have l q ( s ) ≥ ǫ X . Inparticular, either the horizontal or the vertical length of s is ≥ ǫ X / . Geodesics in T (i.e. Teichm¨uller geodesics ) have a simple descriptionin terms of quadratic differentials. If X ∈ T and q is a unit normquadratic differential on X , for t ∈ R define X t ∈ T by the rulethat on a chart of X where q = dz , the chart for X t is x + iy e t/ x + ie − t/ y . Then t X t is a geodesic line determined by X and q and it is parametrized with unit speed. The identity map X → X t is the Teichm¨uller map for these two points in T . There is a naturalquadratic differential q t on X t given by q t = dz in the new charts,and the Teichm¨uller geodesic defined by ( X t , q t ) is the same as theone defined by ( X, q ) except for the reparametrization t t + t . Bydefinition we have q Ht = e t/ q H and q Vt = e − t/ q V . We have a map from the cone g QD ( X ) = QD ( X ) × [0 , ∞ ) /QD ( X ) × { } to T given by ( q, t ) X t with X t described above. Theorem 5.4 (Teichm¨uller’s contractibility theorem) . This map is ahomeomorphism.
For a proof see e.g. [41, Theorem 7.2.1]. A consequence of thetheorem is that any two points in T are joined by a unique Teichm¨ullergeodesic.5. g . Modulus and the Collar Lemma.
The interior of any closedcomplex annulus A is conformally equivalent (or biholomorphic) to theunique flat annulus S × (0 , mπ ) where S is the standard circle with length 2 π . The number m > modulus of A . See Ahlfors’ book [1]for the classical theory. In particular, there is the following equivalentdefinition of the modulus that depends only on the conformal structure,see [1, Chapter I.D, Example 2]: M od ( A ) = sup ρ inf λ ℓ ρ ( λ ) Area ρ ( A )where ρ runs over all conformally equivalent metrics, and λ over allarcs connecting the two boundary components. In particular, if A is an annulus in a complete hyperbolic surface X of finite area andwith the fixed underlying surface Σ, and if the distance between theboundary components is large, then the modulus of A is large. This isbecause we can take the hyperbolic metric for ρ and then the area of A is bounded by the area of X , which in turn is bounded by the topologyof Σ.The following lemma is fundamental for the geometry of hyperbolicsurfaces. See [28, Lemma 13.6], which shows that the distance betweenthe boundary components is large. Lemma 5.5 (The Collar Lemma or the Margulis Lemma) . There arefunctions
F, G : (0 , ∞ ) → (0 , ∞ ) such that lim t → F ( t ) = lim t → G ( t ) = ∞ and such that the following holds. If a hyperbolic surface has a simpleclosed geodesic of length < t , then its F ( t ) -neighborhood is an embeddedannulus whose modulus is ≥ G ( t ) . Projections a . Curve complex; arc and curve complex.
The curve complex C (Σ) is the simplicial complex whose vertex set is the set of curves in S , and simplices are collections of curves that have pairwise disjointrepresentatives. When 6 g − p > δ can be taken to be uniform (e.g. δ = 17) for allsurfaces [2, 23, 25, 39]. For the purpose of this paper, this is however,not important. Theorem 6.1 ([60]) . The 1-skeleton of C (Σ) is a δ -hyperbolic graphwhenever it is connected. It will be more convenient to work with the arc and curve complex AC (Σ). Its vertices are represented by (essential) arcs and curves. Byan arc we mean a path in Σ whose interior points are in Σ r P and whose HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 43 boundary is in P , and it is embedded except possibly at the endpoints.Two arcs are equivalent if they are homotopic through arcs. An arc isessential if it is not homotopic through arcs to a small neighborhood ofa puncture. A simplex in AC (Σ) is a collection of arcs and curves thathave disjoint representatives, except possibly for the endpoints of arcs.The complex AC (Σ) is connected and δ -hyperbolic as soon as 6 g − p >
0. When 6 g − p > C (Σ) ֒ → AC (Σ)is a quasi-isometry. The inverse is constructed by sending an arc α toan essential component of the boundary of the regular neighborhoodof α , see [51, Theorem 1.3].When 6 g − p = 2 (i.e. when (Σ , P ) is the once-punctured torus orthe four times punctured sphere) the complex AC (Σ) is quasiisometricto the Farey graph (hence also hyperbolic), while C (Σ) is an infinitediscrete space.If α, β are two arcs or curves, their intersection number i ( α, β ) isthe smallest cardinality of the intersection of their representatives, notcounting the punctures.We have the following useful estimate on the distance in C (Σ) and AC (Σ). Proposition 6.2. • If α, β are curves and g − p > then d C (Σ) ( α, β ) ≤ i ( α, β ) + 1 . • If α, β are arcs or curves and P = ∅ , then d AC (Σ) ( α, β ) ≤ i ( α, β )+2 .Proof. The first claim is well known; it can be easily proved by induc-tion on the intersection number using surgery. See [21, Lemma 1.1].There are also logarithmic bounds, see [38].The second claim can be proved similarly. E.g. see [39, Definition 3.1and Remark 3.2] for the case when α, β are arcs, when d AC (Σ) ( α, β ) ≤ i ( α, β ) + 1. If α is a curve and β an arc, we can construct an arc α ′ dis-joint from α and with i ( α ′ , β ) ≤ i ( α, β ). Then we have d AC (Σ) ( α, β ) ≤ d AC (Σ) ( α ′ , β ) + 1 ≤ i ( α, β ) + 2. (cid:3) When Σ is a 3 times punctured sphere, the complex AC (Σ) is finite,and is not useful when considering subsurface projections.6. b . Curve complex of the annulus.
When A is an annulus, wedefine C ( A ) = AC ( A ) to be the graph whose vertices are embeddedarcs with endpoints on distinct boundary components of A , moduloisotopy rel boundary, and edges correspond to disjointness. Thus C ( A )is quasi-isometric to Z . c . The Gromov boundary.
Klarreich [50] gave a description ofthe Gromov boundary of the curve complex C (Σ) (or equivalently of AC (Σ)). A point in ∂ C (Σ) is represented by a filling measured geodesiclamination ξ and two such laminations ξ, ξ ′ represent the same point if | ξ | = | ξ ′ | (see Section 5. c ). In other words, a point in ∂ C (Σ) is a fillinggeodesic lamination that admits a transverse measure of full support.We now state Klarreich’s work in more detail. First recall that if x n is a sequence in a δ -hyperbolic space X then after passing to asubsequence one of the following occurs: • x n → z ∈ ∂X , or • there is some x ∈ X so that x n coarsely rotates around x . Thismeans that for any n there is m so that for m > m any geodesic[ x n , x m ] passes within a uniform distance (e.g. 10 δ ) from x .This statement is really an exercise in Gromov products (e.g. thereader should contemplate the case of a locally infinite tree). A moresophisticated approach is via the horofunction boundary, see e.g. [56,Section 3].The theorem of Klarreich can now be summarized as follows. Theorem 6.3 (Klarreich) . There is a coarse map π : T → C (Σ) ∪ ∂ C (Σ) with the following properties.(1) Suppose x n ∈ T , x n → x ∈ T . If π ( x ) ∈ ∂ C (Σ) then π ( x n ) → π ( x ) . If π ( x ) ∈ C (Σ) then π ( x n ) coarsely rotates around π ( x ) .(2) If X ∈ T then π ( X ) is the collection of shortest curves on X (or equivalently, collection of curves of length less than a suitableconstant). If µ ∈ PMF is not filling, π ( µ ) consists of boundarycomponents of the supporting multisurface. If µ is filling then π ( µ ) ∈ ∂ C (Σ) . Moreover, for every b ∈ ∂ C (Σ) the preimage π − ( b ) is nonempty and consists of the simplex of projectivizedtransverse measures on the same underlying foliation. In par-ticular, if µ is uniquely ergodic, the preimage of π ( µ ) is a singlepoint. If A is an annulus, its curve complex is quasi-isometric to Z andthe Gromov boundary has two points. We can think of them as thetwo ways in which a geodesic can spiral in and out of the annulusrepresented by a regular neighborhood of a simple closed geodesic.6. d . Subsurface projections.
This key concept was introduced byMasur and Minsky [61]. Recall our convention about subsurfaces fromSection 5. d . In particular, they are proper and connected. When a sub-surface Y is not an annulus, we define its arc and curve complex AC ( Y )as AC ( ˆ Y ), where ˆ Y is obtained from Y by collapsing each boundary HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 45 component to a puncture. Thus any essential arc with boundary in ∂Y represents a point in AC ( Y ). The complex AC ( Y ) is always δ -hyperbolic and of infinite diameter since Y is not allowed to be a pairof pants.Let Y ⊂ Σ be a connected subsurface different from a pair of pants.Fix a complete hyperbolic metric of finite area on Σ r P and realizeall nonperipheral boundary components of Y by geodesics. If no twoboundary components of Y are parallel, then Y is realized as a totallygeodesic subsurface.Let α be an arc or a curve in Σ, not isotopic into the complementof Y , realized as a geodesic. The intersection Y ∩ α is a curve or acollection of arcs. We define π Y ( α ) ⊂ AC ( Y ) to be this intersection.This is a collection of points in AC ( Y ) at pairwise distance ≤
1, socoarsely the projection is well-defined.If Y has parallel boundary components but is not an annulus (i.e.when a complementary component is an annulus) consider the coveringspace Σ Y → Σ corresponding to Y ⊂ Σ. The subsurface Y lifts to Σ Y and there is a unique representative, up to isotopy, which is totally ge-odesic, and we will identify it with Y . The entire covering space Σ Y isobtained from Y by attaching half-open annuli to the boundary com-ponents. Each annulus is of the form H/ Z , where H is the hyperbolichalf-plane and Z acts by translation along the boundary. The Gromovcompactification of H/ Z is a (compact) annulus, and attaching theseannuli to Y produces a surface Σ Y homeomorphic to Y , with homeo-morphism being canonical up to isotopy. Now define π Y ( α ) ⊂ AC ( Y )as the intersection of the preimage of α in Σ Y with Y . Equivalently,identifying Y with Σ Y , take the closure of the preimage of α , and dis-card the inessential components. The resulting finite collection of arcs(or a curve) is the projection.When Y is the annulus, it is the latter description of the projectionthat generalizes. Namely, Σ Y is an annulus. Again take the closure ofthe preimage of α , and discard the inessential components to get theprojection.We make the same definition when α is a collection of pairwise dis-joint arcs or curves and at least one is not isotopic into the complementof Y .Subsurface projections can also be defined for other subsurfaces andfor geodesic laminations.If Y ′ ⊂ Σ is another subsurface, define π Y ( Y ′ ) = π Y ( ∂Y ′ )assuming the latter is defined; otherwise π Y ( Y ′ ) is undefined. e . Projecting geodesic laminations.
We now define π Y ( ξ ) = π Y ( | ξ | )when ξ is a measured geodesic lamination with support | ξ | . As sug-gested by the notation, it will depend only on the support, and it willbe defined whenever Y ∩ Supp ( ξ ) = ∅ (even after isotopy). If Y is acomponent of Supp ( ξ ) and it is not an annulus we define π Y ( ξ ) to bethe point at infinity of AC ( Y ) represented by | ξ | .If Y is an annulus component of Supp ( ξ ) we define π Y ( ξ ) to be thetwo points at infinity in the curve complex.Now suppose that Y is not a component of Supp ( ξ ). First assumethat Y is realized as a totally geodesic subsurface of Σ. Then | ξ | ∩ Y is the union of a collection of arcs (typically uncountably many,but there are only finitely many isotopy classes) and the underlyingset of a measured geodesic lamination µ . Define π Y ( ξ ) as the set ofthese arcs and boundary components of Supp ( µ ) that are not boundarycomponents of Y .More generally, if Y is not an annulus, we lift to the cover Σ Y andintersect with the totally geodesic copy of Y .Finally, if Y is an annulus, it is crossed by some leaves of ξ . Liftthose leaves to Σ Y and take their closure in Σ Y to get π Y ( ξ ).Note that the set PMF ( Y ) ⊂ PMF consisting of measured geodesiclaminations ξ such that π Y ( ξ ) is defined is open. This follows fromProposition 5.2.6. f . Projection distance.
Let α, β be two curves, or arcs, or subsur-faces, or Riemann surfaces, or measured foliations, so that the projec-tions π Y ( α ) and π Y ( β ) to a subsurface Y are defined. Then define the projection distance d πY ( α, β ) = diam( π Y ( α ) ∪ π Y ( β ))When we use this notation at most one of π Y ( α ) , π Y ( β ) will representa point (or points) at infinity, and in this case the projection distanceis infinite. In all other cases, since the diameter of π Y ( α ) is uniformlybounded, the projection distance is a well-defined finite number.The following triangle inequality is obvious. Proposition 6.4. If π Y ( α ) , π Y ( β ) , π Y ( γ ) are all defined, then d πY ( α, β ) + d πY ( β, γ ) ≥ d πY ( α, γ )The following key inequality was proved by Behrstock. We say thattwo subsurfaces Y, Y ′ overlap if ∂Y ∩ ∂Y ′ = ∅ . Proposition 6.5 ([13, Theorem 4.3]) . There is a constant C such thatthe following holds. Let Y, Z be two overlapping subsurfaces of Σ (i.e. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 47 ∂Y ∩ ∂Z = ∅ , even after isotopy) and let α be a collection of pairwisedisjoint arcs and curves. Assume π Y ( α ) and π Z ( α ) are defined. Then d πY ( α, ∂Z ) ≥ C = ⇒ d πZ ( α, ∂Y ) ≤ C The same statement holds when α is replaced by a foliation. Leininger supplied explicit constant C = 10 and a simple argument,see [57, Lemma 2.13]. When α is replaced by a foliation ξ the proof isan easy consequence, as follows: • Leininger’s proof works with no change if some leaf of ξ intersects ∂Y or ∂Z (in particular, this occurs if ξ is filling). • If ξ is not filling let α = ∂Supp ( ξ ). If π Y ( α ) and π Z ( α ) are defined,the claim about ξ follows (after increasing C by 1) after observingthat d πY ( α, ξ ) ≤ d πZ ( α, ξ ) ≤ • If Y is a component of Supp ( ξ ), then d πZ ( ξ, ∂Y ) ≤
1, and similarlyfor Z .The following was first proved in [61]. A streamlined proof with theexplicit bound is in [14, Lemma 5.3]. Proposition 6.6.
For any two subsurfaces
Y, Z there are only finitelymany subsurfaces W such that d πW ( Y, Z ) > . g . The Bounded Geodesic Image Theorem.
This theorem wasproved by Masur and Minsky [61]. A more combinatorial proof with auniform bound on M was given by Webb [72]. Theorem 6.7.
There exists M = M (Σ) such that the following holds.Let Y ⊂ Σ be a subsurface and g = x , x , · · · , x n a geodesic in C (Σ) such that π Y ( x i ) is defined for all i . Then d πY ( x , x n ) ≤ M . Proposition 6.8.
There is a constant N = N (Σ) so that the followingholds. Suppose π Y ( ν ) is defined and let α ∈ AC ( Y ) and Θ ∈ [0 , ∞ ) such that d πY ( α, ν ) ≥ Θ . Then there is a neighborhood U of ν in PMF such that π Y ( µ ) is defined and in addition d πY ( α, µ ) > Θ − N for all µ ∈ U .Proof. We already noted that π Y is defined on an open subset of PMF .Let µ i → ν in PMF .We first consider the case when Y is a component of Supp ( ν ). If Y is also a component of Supp ( µ i ) there is nothing to prove. Otherwise,the projection π Y ( µ i ) consists of a collection of curves (coming fromboundary components of the support of µ i ) and of a collection of arcs.Denote by λ i either one of the curves in the collection, or one of the essential nonperipheral boundary components of I ∪ ∂Y where I is oneof the arcs in the collection. We view λ i as a measured lamination.and we may assume λ i → λ ∈ PMF . Then λ is supported in Y andsatisfies i ( λ, ν ) = 0. Since π Y ( ν ) fills Y it follows that | λ | = | π Y ( ν ) | .By Theorem 6.3 it follows that d πY ( α, λ i ) → ∞ , so we are done in thiscase since d πY ( λ i , µ i ) ≤
5, see Proposition 6.2. When Y is an annuluswe cannot use Theorem 6.3, but argue directly as follows. Let α bethe closed geodesic homotopic into Y . If µ i has α as a closed leaf then d πY ( α, µ i ) = ∞ so we are done. Otherwise, µ i has leaves that intersect α at a small angle by Proposition 5.2 and then again the projectionsgo to infinity.Second, consider the case when a leaf ℓ of | ν | intersects ∂Y . ByProposition 5.2 there is a leaf ℓ i of µ i for large i that also intersects ∂Y and an arc component of ℓ i ∩ Y is isotopic to an arc component of ℓ ∩ Y . Thus in this case d πY ( µ i , ν ) ≤ Y is an annulus we only get d πY ( µ i , ν ) ≤ N > Z of Supp ( ν ) is isotopic into Y ,but is not Y . By the same argument as in the first case, we see that π Z ( µ i ) go to infinity in AC ( Z ) (again argue directly if Z is an annulus).After passing to a subsequence, we may assume that for i < j we havethat d πZ ( µ i , µ j ) is large. By the Bounded Geodesic Image Theorem 6.7,we see that any geodesic joining π Y ( µ i ) and π Y ( µ j ) must contain acurve disjoint from Z , and so is within distance 1 from π Y ( ν ). Thenby δ -hyperbolic geometry we deduce that d πY ( α, µ i ) < Θ − − δ for atmost one i . So N > δ works in this case. (cid:3) h . Partitioning the subsurfaces and the color preserving sub-group.
We will need the following fact.
Proposition 6.9 ([14, Proposition 5.8]) . The set of subsurfaces of Σ which are not pairs of pants can be written as a finite disjoint union Y ⊔ Y ⊔ · · · ⊔ Y k so that any two subsurfaces in any Y i overlap, and there is a subgroup G <
Mod(Σ) of finite index that preserves each Y i . The subgroup G is the color preserving subgroup . We can furtherarrange that each Y i is a G -orbit. We remark that unlike in [14], { Σ } is not one of the Y i since we are considering only proper subsurfaces.6. i . Large intersection number implies large projection.
No-tice that one can have two curves with large intersection number thatare at distance 2 in the curve complex. Thus the literal converse to
HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 49
Proposition 6.2 does not hold. The following is the correct converse toProposition 6.2.
Lemma 6.10. [71, Theorem 1.5]
For every M there is I such that the following holds. If α, β ∈ S and i ( α, β ) ≥ I then there is a subsurface Y ⊂ Σ such that d πY ( α, β ) ≥ M (where we also allow Y = Σ ).The same statement holds for pairs of arcs-or-curves in AC (Σ) . Verification of the Flow Axioms
Let c be a geodesic (segment, ray or line) in a metric space X . Wewrite ρ c for the nearest point projection from X to Im( c ). Definition 7.1.
The geodesic c is D -contracting for D ≥ B ⊂ X with B ∩ Im( c ) = ∅ the set ρ c ( B ) := ∪ b ∈ B ρ c ( b ) ⊂ Im( c )has diameter ≤ D .For example, if X is δ -hyperbolic then every geodesic is 10 δ -con-tracting. It is an interesting phenomenon that many spaces of inter-est, even though they are not hyperbolic, contain many contractinggeodesics. These are thought of as “hyperbolic directions”. Note thata line in R is not contracting. Definition 7.2.
A geodesic (segment, ray or line) c in a metric space is Morse if for every
A, B there is N = N ( A, B ) so that any (
A, B )-quasi-geodesic c ′ with endpoints in c is contained in the N -neighborhood B N ( c ) of c .The following lemma is well known. The first part states that con-tracting geodesics are Morse, and the second is a variant for geodesiclines. Lemma 7.3. (i) For every
D, A, B there is N = N ( D, A, B ) suchthat every ( A, B ) -quasi-geodesic with endpoints on a D -con-tracting geodesic c is contained in the N -neighborhood of c .(ii) Suppose c is a contracting geodesic line and c ′ is a geodesic linecontained in some neighborhood B M ( c ) of c . Then c ′ is con-tained in the R -neighborhood of c , where R is a function of thecontracting constant D . For the first statement, see e.g. [3]. The idea is that if c ′ containsa long segment outside a big neighborhood of c , then covering thissegment with slightly overlapping big balls that miss c and projecting we see that the projection of the segment to c has much smaller length,and this contradicts the assumption that c ′ is a quasi-geodesic. Thesecond statement can be proved similarly.We will need the following facts about Teichm¨uller geodesics. Recallthat π : T → C (Σ) is the coarse projection to the curve complex.
Proposition 7.4. (1) (Arzela-Ascoli) Any sequence of Teichm¨ullergeodesics that intersect a fixed compact set has a subsequencethat (after reparametrization via translation) converges to a Teich-m¨uller geodesic.(2) If c is a Teichm¨uller geodesic (segment, ray or line) in the thickpart T ≥ ǫ then • πc is a ( K, L ) -quasi-geodesic in C (Σ) , where K, L dependonly on ǫ . For a much more general statement see [67] . • (Minsky [63] ) c is D -contracting for D = D ( ǫ ) . • (the Masur Criterion [59] ) c ( t ) converges as t → ∞ to c ( ∞ ) ∈PMF which is filling and uniquely ergodic and equals thevertical foliation of c . In particular, note that as a consequence of Theorem 6.3 and theMasur Criterion, if c, d are two Teichm¨uller rays in a thick part suchthat their images πc and πd fellow travel in C (Σ) then the verticalfoliations of c and d determine the same point in PMF , and this pointis c ( ∞ ) = d ( ∞ ). Lemma 7.5.
Suppose [ Z n , Y n ] are Teichm¨uller geodesic segments thatconverge to a Teichm¨uller ray c , so that Z n → c (0) . If c and all [ Z n , Y n ] are in a thick part T ≥ ǫ then Y n → c ( ∞ ) .Proof. If this fails, then, after a subsequence, Y n → y = c ( ∞ ). The-orem 6.3 and the Masur criterion imply that π ( y ) = π ( c ( ∞ )) as well.There are two cases.Suppose first π ( y ) ∈ ∂ C (Σ). By Proposition 7.4 πc is a quasi-geodesic. Choose a quasi-geodesic from π ( c (0)) to π ( y ). These twoquasi-geodesics start at the same point and go to distinct boundarypoints, so they coarsely form a tripod, with the center point w ∈ C (Σ),say. By Proposition 7.4, we can choose T >> T in T ≥ ǫ projects in C (Σ) with end-points at distance >> d ( πc (0) , w ). For large n choose W n ∈ [ Z n , Y n ] atdistance T from Z n . Then π ( Y n ) → π ( y ) ∈ ∂ C (Σ) implies that π ( W n )is in a bounded neighborhood of a quasi-geodesic from w to π ( y ). But W n → c ( T ), so for large n , π ( W n ) is in a bounded neighborhood of aquasi-geodesic from w to π ( c ( ∞ )). This is impossible since π ( W n ) isalso far from w . HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 51
The other case is that π ( y ) ∈ C (Σ). Then Theorem 6.3 implies that π ( Y n ) coarsely rotate around π ( y ). Again choose W n ∈ [ Z n , Y n ] at afixed distance T from Z n so that π ( W n ) is much further from π ( Z n )than π ( y ). Thus π ( W n ) also coarsely rotates around π ( y ). But on theother hand W n → c ( T ), so π ( W n ) will stay in an R -neighborhood of π ( c ( T )). Here R is independent of T , so for large T , the π ( W n ) cannotcoarsely rotate. Contradiction. (cid:3) We can now prove that the collection of Teichm¨uller rays that stayin a fixed thick part T ≥ ε satisfies our flow axioms (F1) to (F3). Proof of (F1).
Let c n be Teichm¨uller rays in T ≥ ε . Let Y n := c n (0)and by assumption there are Z n ∈ c n that stay in a bounded sub-set. Let X n ∈ T such that d ( X n , Y n ) is bounded. We need to show Y n → ξ ∈ PMF if and only if X n → ξ ∈ PMF . In both cases d ( Z n , Y n ) , d ( Z n , X n ) → ∞ . We proceed by contradiction. As PMF iscompact, we can, after a subsequence, assume Y n → ξ and X n → ξ ′ with ξ = ξ ′ . Passing to a further subsequence, we may assume that[ Z n , Y n ] converges to a geodesic ray c in T ≥ ǫ . Likewise we may assumethat [ Z n , X n ] converges to a geodesic ray c ′ with c ′ (0) = c (0). ByProposition 7.4 the [ Z n , Y n ] are D -contracting. Lemma 7.3(i) impliesthat [ Z n , X n ] is contained in a fixed neighborhood of [ Z n , Y n ]. Thus c ′ is in a neighborhood of c and the projections of c and c ′ fellow traveland therefore converge to the same point in ∂ C (Σ). By unique ergod-icity (Theorem 6.3 and Proposition 7.4) it follows that c ( ∞ ) = c ′ ( ∞ ).Lemma 7.5 now implies that both Y n and X n converge to this point. (cid:3) Proof of (F2).
Let X ∈ T , ξ + ∈ PMF . Let c be the Teichm¨uller raystarting in X with vertical foliation ξ + . By Theorem 5.4 the map thatassociates to ξ ∈ PMF the evaluation at t of the geodesic ray startingin X with vertical foliation ξ is continuous. Thus for a given t there is aneighborhood U + of ξ + such that d ( c ′ ( t ) , c ( t )) < c ′ (0) = X and the vertical foliation of c ′ is in U + . Let c be a Teichm¨uller ray in T ≥ ε with d ( c (0) , X ) < ρ and c ( ∞ ) ∈ U + . By the Masur criterion (Propo-sition 7.4), c ( n ) → c ( ∞ ). We will show that d ( c ( t ) , c ( t )) is bounded,where the bound depends only on ε and ρ . By Proposition 7.4 theray c is contracting. By Lemma 7.3(i), the geodesic segments [ X, c ( n )]stay in a bounded neighborhood of c . Let c ′ be the limiting ray of (asubsequence of) the [ X, c ( n )]. The [ X, c ( n )] and c ′ stay in the thickpart (as they are in a bounded neighborhood of T ≥ ε ). Thus Lemma 7.5applies and c ( n ) → c ′ ( ∞ ). Therefore c ′ ( ∞ ) = c ( ∞ ). By the Masurcriterion (Proposition 7.4) the vertical foliation of c ′ is c ′ ( ∞ ) = c ( ∞ )and therefore in U + . Thus d ( c ′ ( t ) , c ( t )) <
1. As c ′ stays in a bounded neighborhood of c , d ( c ( t ) , c ′ ( t )) is bounded as well. Thus d ( c ( t ) , c ( t ))is bounded. (cid:3) Remark . The question when Teichm¨uller rays are parallel, or as-ymptotic, is well understood. See [26, 44, 52, 58, 68].
Proof of (F3).
Let X ∈ T K,ρ ( ξ − , ξ + ). Choose a sequence c n as in thedefinition. After a subsequence and reparametrization, c n → c . Thus c is a Teichm¨uller geodesic in T ≥ ǫ that passes within ρ of X . It re-mains to prove that if c, c ′ are two geodesics in the thick part and c ( ±∞ ) = c ′ ( ±∞ ), then c, c ′ are in uniform neighborhoods of eachother. By Lemma 7.3(ii), uniformity is automatic if we can showthat c, c ′ are parallel, i.e. contained in each other’s metric neighbor-hoods. Consider the geodesics [ c (0) , c ′ ( t )], t ≥
0. Let ˜ c be the lim-iting ray of a subsequence. By Lemma 7.3(i) the [ c (0) , c ′ ( t )] and ˜ c stay in a bounded neighborhood of c ′ (where the bound depends on d ( c (0) , c ′ (0))) and in particular in some thick part. Thus Lemma 7.5yields ˜ c (+ ∞ ) = c ′ (+ ∞ ) = c (+ ∞ ). By the Masur criterion (Propo-sition 7.4) the vertical foliations of c and ˜ c are ˜ c (+ ∞ ) and c ( ∞ ) andtherefore coincide. Of course ˜ c (0) = c (0). Thus the restriction of c to[0 , + ∞ ) coincides with ˜ c and stays in a bounded neighborhood of c ′ .Similarly, the restriction of c to ( −∞ ,
0] stays in a bounded neighbor-hood of c ′ . (cid:3) Remark . The Gardiner-Masur theorem [32] states that if ξ ± aretwo measured foliations such that i ( ξ + , µ ) + i ( ξ − , µ ) > µ ∈ MF , then there is a unique Teichm¨uller geodesic with verticaland horizontal foliations ξ ± . Conversely, any pair ξ ± of a horizontaland vertical foliation satisfies this condition. Remark . We stated the axioms with an eye towards applying themto
Out ( F n ). At present we do not know how to make them all work, butwe point out the following. For R >> R -thick part of Culler-Vogtmann’s Outerspace and whose projections to the complex of free factors are ( R, R )-quasi-geodesics. The main ingredients needed for the flow axioms areall known in the
Out ( F n ) context: these lines are contracting [16], theMasur Criterion holds [65], and the analog of Klarreich’s theorem is in[18, 36]. The projection axioms are more subtle, but see [17].8. Teichm¨uller geodesics intersecting the thin part a . Thick or thin.
Suppose X ∈ T ≥ ǫ and ξ ∈ PMF . Section 7was concerned with Teichm¨uller rays completely contained in the thickpart. Here we want to establish that if there are short curves along the HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 53 ray, then there is a (proper) subsurface where the projection distancebetween the vertical foliation and a suitable point thought of as theprojection of the initial point of the ray, is large. The next theorem isa variant of a theorem of Rafi [66].
Theorem 8.1.
For each collection Y i (see Section 6. h ) fix a basepoint X i ∈ Y i , and also fix X ∈ T . Then for every Θ > there is a compactset K ⊂ T such that for any ( g, ξ ) ∈ G × PMF one of the followingholds:(a) The Teichm¨uller ray c starting in gX with vertical foliation ξ is contained in G · K , or(b) there is some Y i and Z ∈ Y i so that d πZ ( gX i , ξ ) > Θ . In order to prove this theorem we will need the following statement.
Proposition 8.2.
Fix a Riemann surface X and a finite filling col-lection of curves β i in X . For every k there exists ǫ > so that thefollowing holds. Suppose g is a Teichm¨uller geodesic with g = X andwith vertical measured foliation ξ + at X . Assume that ξ + is filling.Suppose there is a curve α of hyperbolic length ǫ at g T for some T > ,and let β = β i be one of the curves that intersects α . Then there is acomponent Y of X cut open along α , so that after putting Y , β and theleaves of ξ + in minimal position, there are arcs β ′ , ℓ of intersections of β and ξ + with Y so that the intersection number i Y ( β ′ , ℓ ) > k .Proof of Theorem 8.1 assuming Proposition 8.2. If ξ is not filling wecan take Y to be a component of Supp ( ξ ) and then (b) holds since d πY ( gX i , ξ ) = ∞ . So assume ξ is filling. By equivariance, we can alsoassume g = 1. Fix a finite collection of curves β , · · · , β m that fill Σ.Note that there is a bound on the intersection numbers between any β j and any ∂X i . This means that when d πY ( β j , X i ) is defined, it isuniformly bounded.Now suppose that there is a curve s that has length < ǫ somewherealong c . Choose β = β j that intersects s . By Proposition 8.2 there isa subsurface Y of Σ and arcs β ′ , ℓ of intersection between β, ξ with Y whose intersection number is as large as we want, if ǫ is small enough.By Lemma 6.10 there is a further subsurface Z of Y such that d πZ ( β, ξ ) is large. Thus Z satisfies the conclusion, since replacing β by X i (where Z ∈ Y i ) changes projection distance by a bounded amount.If no curve gets ǫ -short along c , then (a) holds with a suitable K (seeTheorem 5.1). (cid:3) b . Outline for Proposition 8.2.
We will follow Rafi’s strategy in[66] where he proved similar statements. Here is the basic idea. Since α is short in g T it will have, by the Collar Lemma 5.5, an annulusneighborhood where the distance between the boundary components ismuch larger than the length of α . Thus the arcs of intersection between β and X r α must be long, as they have to go across the annulus. Onthe other hand, if the intersection numbers are bounded, we will arguethat the horizontal lengths of these arcs with respect to the quadraticdifferential q T are too small. Since the vertical lengths are small as well,given that they are bounded at time 0, this will give a contradiction.Technically, we will work with the metric l q T induced by q T and needto review the work of Minsky [62] on the Collar Lemma in this setting.8. c . Primitive annuli.
Let X be a Riemann surface and q a quadraticdifferential of area 1 on X . Thus X is equipped with the associatedflat metric l q (with cone type singularities). The following conceptwas introduced by Minsky [62, Section 4.3]. Consider a closed annulus B ⊂ X with piecewise smooth boundary curves ∂ and ∂ . We say that B is regular if • ∂ and ∂ are equidistant from each other, • either ∂ is a geodesic, or else the curvature vector of ∂ pointsaway from B (and the internal angles are ≥ π ), and likewise either ∂ is a geodesic or the curvature vector of ∂ points into B (and theinternal angles are ≤ π ).Denote by κ ( ∂ i ) the total signed curvature of ∂ i . Thus κ ( ∂ ) ≤ κ ( ∂ ) ≥
0. These numbers can also be thought of as the total turningnumber of ∂ i with respect to the horizontal (or vertical) foliation. Wecall the annulus B primitive if it is regular and contains no singularpoints in its interior. In that case κ ( ∂ ) = − κ ( ∂ ) is a nonnegativeintegral multiple of π , and we denote this quantity by κ ( B ). When κ ( B ) = 0 then B is a flat cylinder.Other than the flat cylinder (when κ = 0) an example of a primitiveannulus (with κ = 2 π ) is the region in the plane between two concentriccircles (here ∂ is the smaller circle). The following is a Theorem ofMinsky [62, Theorem 4.6]. Proposition 8.3.
For any topological surface Σ there are constants m , c , c > so that for any Riemann surface X homeomorphic to Σ and any homotopically nontrivial annulus A ⊂ X with M od ( A ) ≥ m there exists a primitive annulus B ⊂ A homotopic to A and with M od ( B ) ≥ c M od ( A ) − c .Remark . In the statement of [62, Theorem 4.6] there is a typo; theinequality in
M od ( A ) ≥ m is reversed. HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 55
Further, [62, Theorem 4.6] finds only a regular annulus B . One canthen pass to a primitive subannulus by taking a parametrization bycurves equidistant from ∂ and passing to a subannulus bounded bytwo such curves that avoids the singular points. Since the number ofsingular points is bounded by the topology of X , we can arrange thatthe modulus still satisfies the above estimate. See [62, Theorem 4.5]and the paragraph before it.We will need the following elementary facts about a primitive annulus B . Denote by d the distance between the two boundary components. Proposition 8.5.
Let B ⊂ X be an essential primitive annulus.(i) If κ ( B ) = 0 then M od ( B ) = dl q ( ∂ ) ,(ii) If κ ( B ) = nπ > then κ ( B ) M od ( B ) ≤ c log (cid:18) dl q ( ∂ ) (cid:19) + c for constants c , c > that depend only on the topology of thesurface Σ underlying X .(iii) Area ( B ) = dl q ( ∂ ) + κ ( B ) d .Proof. (i) is the definition of the modulus of a flat cyclinder. (ii) is[66, Lemma 3.6]. For (iii), we compute the area by integrating lengthsof equidistant curves. The curve at distance r from ∂ has length l q ( ∂ ) + κ ( B ) r so Area ( B ) = Z d ( l q ( ∂ ) + κ ( B ) r ) dr = d l q ( ∂ ) + κ ( B ) d (cid:3) Proof of Proposition 8.2.
For this proof the reader is invited to reviewSection 5, and particularly subsections 5. f and 5. g We will be workingwith singular Euclidean metrics l q t associated to quadratic differen-tials q t along g t . Let h t ( α ) be the horizontal length of α at time t .By the Collar Lemma, X contains an annulus with core curve α andwith arbitrarily large modulus, if ǫ is sufficiently small. By the aboveProposition, ( g T , l q T ) contains a primitive annulus B with arbitrarilylarge modulus and core curve homotopic to α . Instead of cutting along α we will cut along ∂ (which is homotopic to α ). If ∂ is separating,we let Y be the component of X r ∂ that contains B . If ∂ is non-separating, we let Y = X r ∂ and note that the core curve of B ishomotopic to one of the two copies of ∂ in Y .Let d be the distance between ∂ and ∂ . Then dl qT ( ∂ ) is arbitrarilylarge, by Proposition 8.5 (i) and (ii). Since the area of B is ≤ the area of X is 1) it follows from Proposition 8.5 (iii) that l q T ( ∂ ) isarbitrarily small (can be made arbitrarily close to 0 if ǫ is sufficientlysmall). Thus h T ( α ) ≤ l q T ( α ) ≤ l q T ( ∂ ) is also arbitrarily small. Sincethe horizontal length is growing, we have that h t ( α ) is arbitrarily smallfor t ∈ [0 , T ]. Put α, β, ∂Y and leaves of ξ + in minimal position withrespect to each other. The vertical foliation restricted to Y induces adecomposition of Y into strips, each foliated by pairwise isotopic arcsof intersection between Y and the leaves of ξ + . See Figure 3. Figure 3.
Singular leaves, drawn in red, decompose thesubsurface Y (disk with three holes) into (six) strips, eachfoliated by arcs of ξ + , some of which are drawn in blue;one of the strips is shaded in gray.Each such strip has a width, i.e. the transverse measure assignedto a cross section by the measure on ξ + . The sum of the widths overall the strips is h t ( α ) (when α is nonseparating) or h t ( α ) (when α isseparating). Assuming the conclusion is false, each arc of β intersectseach strip at most k times, so its horizontal length is at most k h t ( α ).The vertical length of β at g = X is bounded above by some fixednumber, say L . Likewise, the vertical length of α at g = X is bounded below by some fixed constant K depending only on X (see Lemma 5.3,this is because the horizontal length is very small and the q -length of α is bounded below). Since the ratio between the vertical lengths of β and α does not change over time, it follows that the ratio l q T ( β ′ ) /l q T ( α )is bounded by a function of k for any arc β ′ . The ratio l q T ( β ′ ) /l q T ( ∂ )is even smaller, and is thus also bounded. It follows that no such arcis long enough to traverse the annulus B , as soon as d/l q T ( ∂ ) is largeenough. Thus β cannot intersect α . Contradiction. (cid:3) HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 57 The Farrell-Jones Conjecture for mapping classgroups
In this section we prove that mapping class groups satisfy the Farrell-Jones Conjecture. We fix a closed oriented surface Σ of genus g with p punctures and write Mod(Σ) for its mapping class group.We start with a proof of Theorem B, which we restate for conve-nience. Theorem B.
Let Σ be a closed oriented surface of genus g with p punctures. Assume that g +2 p − > . Then the action of its mappingclass group Mod(Σ) on the space
PMF of projective measured foliationson Σ is finitely F -amenable where F is the family of subgroups that areeither virtually cyclic or virtually fix a subsurface of Σ . Earlier we described F as the family of subgroups that virtuallyfix a curve or are virtually cyclic. This is the same family, becausea subgroup that fixes a subsurface fixes all boundary curves of thesubsurface virtually, and if a subgroup fixes a curve, then it also fixesthe corresponding annulus. Proof of Theorem B.
By Proposition 4.4 it suffices to show that theaction of the color preserving subgroup
G <
Mod(Σ) (see Section 6. h )is finitely F -amenable. Let T = T be the Teichm¨uller space of Σ andlet ∆ = PMF . Thus T = T ∪ ∆ is homeomorphic to a ball and T isits interior. To finish defining the flow data, for a compact set K ⊂ T let G K be the set of Teichm¨uller rays which are contained in G · K . Bythe Masur Criterion (see Proposition 7.4) every such ray converges toa point of ∆.We now define the projection data. For a subsurface Y ⊂ Σ let∆( Y ) ⊂ ∆ be the open set of foliations whose support is not disjointfrom Y . The collection of subsurfaces which are not pairs of pants ispartitioned into subcollections Y , · · · , Y k each of which is a G -orbitand consists of overlapping subsurfaces (i.e. Y, Y ′ ∈ Y i , Y = Y ′ implies ∂Y ∩ ∂Y ′ = ∅ ). We set Y = { Y , · · · , Y k } . Projections and projectiondistance d πY ( X, Z ) was defined in Sections 6. d and 6. f and d πY ( X, ξ ) for ξ ∈ ∆( Y ) in Sections 6. e and 6. f . Axiom (P1) is obvious, (P2) wasrecorded as Proposition 6.4, (P3) is Proposition 6.5, (P4) is Proposition6.6 and (P5) is Proposition 6.8. This finishes the projection axioms.Axiom 1.5 holds by Theorem 8.1 and the flow axioms were provedin Section 7. (cid:3) Corollary 9.1.
Assume g + 2 p − > . Then the action of theMapping class group Mod(Σ) on the Thurston compactification T = T ∪ PMF of Teichm¨uller space is finitely F -amenable where F is thefamily of subgroups that are either virtually cyclic or stabilize a subsur-face.Proof. By Theorem B the action of Mod(Σ) on
PMF is N - F -amenablefor some N . Let N ′ := dim T . We will show that the action of Mod(Σ)on T is N + N ′ + 1- F -amenable.Let S ⊆ Mod(Σ) be finite. Then there exists an open F -cover ofthe product Mod(Σ) ×PMF of order at most N that is S -long in thegroup coordinate. It is not difficult to extend this cover to a Mod(Σ)-invariant collection U of open F -subsets of Mod(Σ) ×T of order atmost N satisfying ∀ ( g, ξ ) ∈ Mod(Σ) ×PMF ∃ U ∈ U with gS ×{ ξ } ⊆ U, see Lemma 3.16. Teichm¨uller space T carries the structure of a properMod(Σ)- CW -complex. (Even simpler, we could restrict here to a finiteindex subgroup of Mod(Σ) that acts freely.) We therefore find an openFin-cover V of T of dimension N , where Fin denotes the family of finitesubgroups. Now the cover of Mod(Σ) ×PMF consisting of all U ∈ U and all sets of the form Mod(Σ) × V with V ∈ V is an open F cover, isof order at most N + N ′ + 1, and is S -long in the group coordinate. (cid:3) We now write c (Σ) := 6 g + 2 p −
6. For a subsurface Y of Σ we write Y ′ for the closed surface obtained by collapsing the boundary curvesof Y to punctures. Note that c ( Y ′ ) < c (Σ). Lemma 9.2.
Let G be a subgroup of Mod(Σ) that fixes a subsurface Y of Σ . Then there exists a finite index subgroup G of G and a centralextension Z k i −→ G p −→ Q where Q is a subgroup of a product of mapping class groups of closedsurfaces ˆ Y i with punctures for which c ( ˆ Y i ) < c (Σ) .Proof. Let Y := Y and let Y , . . . , Y n be the components of the comple-ment. For each Y i that’s not an annulus let ˆ Y i be the surface obtainedfrom Y i by collapsing the boundary components to punctures. Then c ( ˆ Y i ) < c (Σ). Since G permutes the Y i ’s and their boundary compo-nents, there is a finite index subgroup G of G that preserves each Y i and each boundary component of Y i . Restriction to each non-annular Y i followed by capping off the boundary components induces a grouphomomorphism [28, Proposition 3.19] G p −→ Mod(Σ) × . . . × Mod(Σ) n . HE FARRELL-JONES CONJECTURE FOR MAPPING CLASS GROUPS 59 whose kernel is the free abelian group generated by Dehn twists aroundboundary curves of Y and is central in G . (cid:3) We will now use some of the classes of groups introduced in Sec-tion 4. b . Lemma 9.3.
The mapping class group
Mod(Σ) of Σ belongs to theclass of groups AC ( VNil ) .Proof. We will proceed by induction on c (Σ). In the sporadic case c (Σ) = 0 we have either g = 0, p ≤ g = 1, p = 0. If g = 0, p ≤
3, then Mod(Σ) is finite and belongs to AC ( VNil ). If g = 1, p = 0, then Mod(Σ) = SL ( Z ) acts cocompactly on a locally finitetree T . The geodesic ray compactification T of the tree is a compact ER and the action of SL ( Z ) on T is finitely VCyc-amenable by [8].Alternatively, we could use the action of SL ( Z ) on the hyperbolic plane H as discussed in Example 1.9. Thus Mod(Σ) ∈ AC ( VNil ).Suppose now c (Σ) >
0. The Thurston compactification T (Σ) ofTeichm¨uller space is homeomorphic to a closed Euclidean ball and inparticular a compact ER . Virtually cyclic groups belong to VNil andtherefore also to AC ( VNil ). Lemma 4.9 (b) implies that AC ( VNil ) isclosed under finite products, central extensions with finitely generatedkernel, taking subgroups and with taking overgroups of finite index.Therefore the induction hypothesis and Lemma 9.2 imply that all sta-bilizers of subsurfaces of Σ belong to AC ( VNil ). Now Corollary 9.1allows us to use the defining property of the operation ac . ThereforeMod(Σ) belongs to AC ( VNil ). (cid:3) Proof of Theorem A.
Corollary 4.10 and Proposition 4.11 imply thatall groups in AC ( VNil ) satisfy the Farrell-Jones Conjecture. Sincemapping class groups of surfaces belong to this class by Lemma 9.3they satisfy the Farrell-Jones Conjecture. (cid:3)
Remark . The
Farrell-Jones Conjecture with wreath products [11,Sec. 6] is a strengthening of the Farrell-Jones Conjecture that has theadvantage that it is closed under taking finite index overgroups. Sincethe class AC ( VNil ) is closed under taking finite overgroups and un-der finite products all groups in AC ( VNil ) satisfy the Farrell-JonesConjecture with wreath products. In particular, mapping class groupsof surfaces satisfy the Farrell-Jones Conjecture with wreath products.This is also true for all groups in the class AC ( VSol ). References [1] L. V. Ahlfors.
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