Classifying spaces for families and the Farrell-Jones Conjecture

Classifying spaces for families and the Farrell-Jones Conjecture

Wolfgang Lück Bonn Germany

email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/

Göttingen, November 2018

Classifying spaces for families of subgroups

Definition (G-CW-complex)

AG-CW-complexX is aG-space together with aG-invariant filtration

∅=X−1⊆X₀⊆. . .⊆Xn⊆. . .⊆ [

n≥0

Xn=X

such thatX carries thecolimit topologywith respect to this filtration, andX_nis obtained fromXn−1for eachn≥0 byattaching equivariant n-dimensional cells, i.e., there exists aG-pushout

`

i∈InG/H_i×Sⁿ⁻¹

`

i∈Inq_iⁿ

//

X_n−1

`

i∈InG/H_i×Dⁿ

`

i∈InQⁿ_i

//Xn

Remark (G-CW in terms ofCW)

A G-CW -complex X is the same as a G-CW -complex together with a cellular G-action such that for every open cell e and g ∈G satisfying g·e∩e6=∅we have gx =x for every x ∈e.

Example (Simplicial actions)

LetX be a simplicial complex. Suppose thatGacts simplicially onX. ThenGacts simplicially also on thebarycentric subdivisionX⁰, and the G-spaceX⁰ inherits the structure of aG-CW-complex.

Example (Smooth actions)

LetGact properly and smoothly on a smooth manifoldM.

ThenM inherits the structure of aG-CW-complex.

Definition (ProperG-action)

AG-spaceX is calledproperif for each pair of pointsx andy inX there are open neighbourhoodsV_x ofx andW_y ofy inX such that set {g∈G|gVx ∩Wy 6=∅}is finite.

Lemma

1 A proper G-space has always finite isotropy groups.

2 A G-CW -complex X is proper if and only if all its isotropy groups are compact.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and taking subgroups.

A groupGis calledvirtually cyclicif it is finite or containsZas a subgroup of finite index.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (ClassifyingG-CW-complex for a family of subgroups) LetF be a family of subgroups ofG. A model for theclassifying G-CW-complex for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofEF(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →EF(G).

We abbreviateE G:=EF IN(G)and call it theuniversal G-CW-complex for properG-actions.

We also writeEG=ET R(G).

Theorem (Homotopy characterisation ofEF(G)) LetF be a family of subgroups.

There exists a model for E_F(G);

Two models for EF(G)are G-homotopy equivalent;

A G-CW -complex X is a model for EF(G)if and only if the H-fixed point set X^H is contractible for each H ∈ F and X^H is empty for H ∈ F/ .

Remark ((Another) Homotopy characterisation of EF(G))

Let X be a G-CW -complex whose isotropy groups belong toF. Then X is a model for EF(G)if and only if the two projections X ×X →X to the first and to the second factor are G-homotopic and for each H ∈ F there exists x ∈Gx with H ⊆Gx.

TheG-spaceG/Gis model forEF(G)if and only ifF =ALL.

EG→BG:=G\EGis a model for theuniversalG-principal bundleforG-principal bundles overCW-complexes.

A freeG-CW-complexX is a model forEGif and only ifX/Gis an Eilenberg MacLane space of type(G,1).

Example (Infinite dihedral group)

LetD_∞=Z o Z/2=Z/2∗Z/2 be the infinite dihedral group.

A model forED∞is the universal covering ofRP^∞∨RP^∞. A model forE D∞isRwith the obviousD∞-action.

Remark (Maps between classifying spaces for families) LetF andG be two families of subgroups of G. Then the following assertions are equivalent:

There is a G-map EF(G)→EG(G);

The set[EF(G),EG(G)]^G consists of precisely one element;

The projection EF(G)×EG(G)→EF(G)is a G-homotopy equivalence;

F ⊆ G.

Special models for E G

We want to illustrate that theG-spaceE Goften has very nice geometric models and appears naturally in many interesting situations.

Theorem (Simplicial Model)

LetP∞(G)be the geometric realisation of the full simplicial complex on the set G with the obvious simplicial G-action.

Then its barycentric subdivision is a model for E G.

Theorem

Consider the G-space X_G =

f:G→[0,1]

f has finite support, X

g∈G

f(g) =1

with the topology coming from the supremum norm. It is G-homotopy equivalent to E G.

The spacesX_G andP∞(G)have the same underlying sets but in general they have different topologies.

The identity map induces aG-mapP∞(G)→X_Gwhich is a

G-homotopy equivalence, but in general not aG-homeomorphism.

Theorem (Discrete subgroups of almost connected Lie groups) Let L be a Lie group with finitely many path components. Let G ⊆L be a discrete subgroup of L.

Then L contains a maximal compact subgroup K , which is unique up to conjugation, and L/K with the obvious left G-action is a finite dimensional G-CW -model for E G.

Theorem (Actions on CAT(0)-spaces)

Let X be a proper G-CW -complex. Suppose that X has the structure of a complete simply connectedCAT(0)-space for which G acts by isometries. Then X is a model for E G.

The result above contains as special case proper isometric G-actions onsimply-connected complete Riemannian manifolds with non-positive sectional curvatureand properG-actions on trees.

TheRips complexP_d(G,S)of a groupGwith a symmetric finite setS of generators for a natural numberd is the geometric realisation of the simplicial set whose set ofk-simplices consists of(k+1)-tuples(g₀,g₁, . . .g_k)of pairwise distinct elements g_i ∈Gsatisfyingd_S(g_i,g_j)≤d for alli,j∈ {0,1, . . . ,k}.

The obviousG-action by simplicial automorphisms onP_d(G,S) induces aG-action by simplicial automorphisms on the

barycentric subdivisionP_d(G,S)⁰.

Theorem (Rips complex of a hyperbolic group,Meintrup-Schick) Let G be a discrete group with a finite symmetric set of generators.

Suppose that(G,S)isδ-hyperbolic for the real numberδ >0. Let d be a natural number with d ≥16δ+8.

Then the barycentric subdivision of the Rips complex P_d(G,S)⁰ is a finite G-CW -model for E G.

LetΓ^s_g,r be themapping class groupof an orientable compact surfaceF of genusgwithspunctures andr boundary components.

We will always assume that 2g+s+r >2, or, equivalently, that the Euler characteristic of the punctured surfaceF is negative.

It is well-known that the associatedTeichmüller spaceT_g,r^s is a contractible space on whichΓ^s_g,r acts properly.

Theorem (Teichmüller space)

TheΓ^s_g,r-spaceT_g,r^s is a model for EΓ^s_g,r.

Example (SL2(Z))

In order to illustrate some of the general statements above we consider the special exampleSL₂(Z).

The groupSL₂(R)is a connected Lie group andSO(2)⊆SL₂(R) is a maximal compact subgroup. HenceSL₂(R)/SO(2)is a model forE SL₂(Z).

Since the 2-dimensional hyperbolic spaceH²is a

simply-connected Riemannian manifold, whose sectional

curvature is constant−1 andSL₂(Z)acts proper on it by Moebius transformations, theSL₂(Z)-spaceH²is a model forE SL₂(R).

The groupSL₂(R)acts by isometric diffeomorphisms onH²by Moebius transformations. This action is proper and transitive. The isotropy group ofz =i isSO(2). Hence theSL₂(Z)-spaces SL₂(R)/SO(2)andH²areSL₂(Z)-diffeomorphic.

Example (continued)

The groupSL₂(Z)is isomorphic to the amalgamated product Z/4∗_Z/2Z/6. This implies that there is a tree on whichSL₂(Z) acts with finite stabilisers. The tree has alternately two and three edges emanating from each vertex.

This is a 1-dimensional model forE SL₂(Z).

The tree model and the other model given byH²must be SL₂(Z)-homotopy equivalent. Here is a concrete description of such aSL₂(Z)-homotopy equivalence.

Example (continued)

Divide the Poincaré disk into fundamental domains for the SL₂(Z)-action.

Each fundamental domain is a geodesic triangle with one vertex at infinity, i.e., a vertex on the boundary sphere, and two vertices in the interior.

Then the union of the edges, whose end points lie in the interior of the Poincaré disk, is a treeT withSL₂(Z)-action which is the tree model above.

The tree is aSL₂(Z)-equivariant deformation retraction of the Poincaré disk. A retraction is given by moving a pointpin the Poincaré disk along a geodesic starting at the vertex at infinity, which belongs to the triangle containingp, throughpto the first intersection point of this geodesic withT.

The Farrell-Jones and the Baum-Connes Conjectures

Conjecture (K-theoretic Farrell-Jones Conjecture)

TheK -theoretic Farrell-Jones Conjecturewith coefficients in the ring R for the group G predicts that theassembly map, which is the map induced by the projection E_VCYC(G)→G/G,

H_n^G(EVCYC(G),K_R)→H_n^G(G/G,K_R) =K_n(RG) is bijective for all n∈Z.

H_∗^G(−;K_R)is aG-homology theory satisfying H_n^G(G/H,K_R)∼=K_n(RH)forn∈Z.

There is also anL-theoretic version.

The basic idea is to understand theK-theory ofRGin terms of its values onRV for all virtually cyclic subgroupsV and just reduce the computation for generalGto the virtually cyclic subgroups V ⊆G.

In general the right hand side is the hard part and the left side is the more accessible part, since for equivariant homology theories there are methods for its computations available, for instance spectral sequences and equivariant Chern characters.

Often the assembly maps have a more structural geometric or analytic description, which are more sophisticated and harder to construct, but link the Farrell-Jones Conjecture to interesting problems in geometry, topology, algebra or operator theory and are relevant for proofs.

Conjecture (Baum-Connes Conjecture)

TheBaum-Connes Conjecturewith coefficients in the ring R for the group G predicts that theassembly map, which is the map induced by the projection E G→G/G,

K_n^G(E G,K_R)→K_n^G(G/G,K_R) =K_n(C_r^∗(G)) is bijective for all n∈Z.

K_∗^G(−)is equivariant topological homology. It satisfies K_n^G(G/H)∼=K_n(C_∗^r(H)), whereK_n(C_r^∗(H))is the topological K-theory of the reduced groupC^∗-algebra ofH.

The importance of these conjectures is that they imply prominent conjectures due toBass,Borel,Kaplansky,Novikov,Serre, . . . and have meanwhile been proved for large classes of groups.

Finiteness properties

Finiteness propertiesof the spacesEGandE G have been intensively studied in the literature. We mention a few examples and results.

IfEGhas a finite-dimensional model, the groupGmust be

torsionfree. There are often finite models forE Gfor groupsGwith torsion.

Often geometry provides small models forE Gin cases, where the models forEGare huge. These small models can be useful for computations concerningBGitself.

Theorem (A criterion for 1-dimensional models forBG,Stallings, Swan)

The following statements are equivalent:

There exists a1-dimensional model for EG;

There exists a1-dimensional model for BG;

The cohomological dimension of G is less or equal to one;

G is a free group.

Theorem (A criterion for 1-dimensional models forE G, Dunwoody, Karras-Pietrowsky-Solitar)

There exists a1-dimensional model for E G if and only if the cohomological dimension of G over the rationalsQis less or equal to one.

Suppose that G is finitely generated. Then there exists a 1-dimensional model for E G if and only if B is virtually finitely generated free.

Theorem (Virtual cohomological dimension and dim(E G),Lück) Let G be virtually torsionfree.

Then

vcd(G)≤dim(E G) for any model for E G.

Let l ≥0be an integer such that for any chain of finite subgroups H₀(H₁(. . .(Hr we have r ≤l.

Then there is a model for E G of dimensionmax{3,vcd(G)}+l.

Problem (Brown)

For which groups G, which are virtually torsionfree, does there exist a G-CW -model for E G of dimensionvcd(G)?

The results above do give some evidence for a positive answer.

However,Leary-Nucinkishave constructed groups, where the answer is negative.

They even show that the upper bounds given above are optimal.

Theorem (Leary-Nucinkis)

Let X be a CW -complex. Then there exists a group G with X 'G\E G.

Groups with special maximal finite subgroups

LetMF IN be the subset ofF IN consisting of elements inF IN which are maximal inF IN.

Assume thatGsatisfies the following assertions:

(M) Every non-trivial finite subgroup ofGis contained in a unique maximal finite subgroup;

(NM) M ∈ MF IN,M 6={1} ⇒N_GM =M.

Here are some examples of groupsGwhich satisfy conditions (M) and (NM):

Extensions 1→Zⁿ→G→F →1 for finiteF such that the conjugation action ofF onZⁿis free outside 0∈Zⁿ; Fuchsian groups;

One-relator groupsG.

For such a group there is a nice model forE Gwith as few non-free cells as possible.

Let{(M_i)|i ∈I}be the set of conjugacy classes of maximal finite subgroups ofM_i ⊆G.

By attaching freeG-cells we get an inclusion ofG-CW-complexes j₁: `

i∈IG×_Mi EM_i →EG.

DefineX as theG-pushout

`

i∈IG×_Mi EM_i ^{j1 //}

u₁

EG

f₁

`

i∈IG/M_i ^{k1 //}X

whereu₁is the obviousG-map obtained by collapsing eachEM_i to a point.

Theorem

The G-space X is a model for E G.

This small model is very useful for computation of all kind ofK- andL-groups ofRG, provided that the Farrell-Jones Conjecture is true. These computations have interesting applications to

questions about the classification of manifolds and of certain C^∗-algebras.

The potential of these models is already interesting for ordinary group (co-)homology as illustrated next.

Consider the pushout obtained from theG-pushout above by dividing out theG-action

`

i∈IBM_i ^//

BG

`

i∈Ipt ^//G\E G The associated Mayer-Vietoris sequence yields

. . .→He_p+1(G\E G)→M

i∈I

He_p(BM_i)→He_p(BG)

→He_p(G\E G)→. . .

In particular we obtain an isomorphism forp≥dim(E G) +1 M

i∈I

H_p(M_i)−^∼=→H_p(G).

LetGbe one relator-group. ThenGhas a model forE Gof dimension 2 and contains up to conjugacy precisely one maximal subgroupM. The subgroupM is isomorphic toZ/mfor some m≥1.

Hence we get forn≥3

H_n(Z/m)−^∼=→H_n(G).

Outlook

We have only presented a tip of an iceberg.

There are many applications or appearances of the classifying spaces for families we have not mentioned at all.

They play a prominent role inequivariant homotopy theory.

Their definition makes also sense fortopological groups, where they have prominent appearances in the theory of Lie groups or of reductivep-adic groups.

Often there are nice geometric models forE Galso for topological groups, for instance theBruhat-Tits buildingfor reductivep-adic groups, thespace of Riemannian metricson a closed smooth manifoldMwith its action of the group of selfdiffeomorphisms of M, . . .