Classifying spaces for families (Lecture III)

Classifying spaces for families (Lecture III)

Wolfgang Lück Bonn Germany

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

Oberwolfach, October 2017

Outline

We formulate theFarrell-Jones Conjecture for torsionfree groups and discuss some applications.

We introduce the notion of aclassifying space for a family of subgroupsand explain its relevance for group theory and topology.

We present somenice modelsforE G.

We discussfiniteness propertiesofEG,E GandE G.

The Farrell-Jones Conjectures for torsionfree groups

Recall the following formulas for a regular ringRand a generalized homology theoryH_∗, which look similar:

Kn(R[Z]) ∼= Kn(R)⊕Kn−1(R);

H_n(BZ) ∼= H_n(pt)⊕ H_n−1(pt).

IfGandK are groups, then we have the following formulas, which look similar:

Ke_n(Z[G∗K]) ∼= Ke_n(ZG)⊕Ke_n(ZK);

He_n(B(G∗K)) ∼= He_n(BG)⊕He_n(BK).

This raises the question whether there is a generalized homology theoryH_∗ such thatH_n(BG)∼=K_n(RG)holds forn∈Z.

If this is true, we would get for the trivial group such that H_n(pt)∼=K_n(R)holds forn∈Z.

Notice that there is a spectrumK_Rsatisfyingπn(K_R)∼=Kn(R)for n∈Zand for any spectrumEthere is a generalized homology theoryH_∗(−,E)satisfyingH_n(pt;E)∼=πn(E)forn∈Z.

Hence the obvious candidate forH_∗ isH∗(−;K_R).

Moreover, there exists a natural mapHn(BG;K_R)→Kn(RG), calledassembly map.

Conjecture (K-theoretic Farrell-Jones Conjecture for torsionfree groups and regular rings)

TheK -theoretic Farrell-Jones Conjecturewith coefficients in the regular ring R for the torsionfree group G predicts that theassembly map

H_n(BG;K_R)→K_n(RG) is bijective for every n∈Z.

The condition thatRis regular is necessary. Recall that the Bass-Heller-Swan Theorem gives an isomorphism

K_n(RZ)∼=K_n(R)⊕K_n−1(R)⊕NK_n(R)⊕NK_n(R) whereas the conjecture above predicts

Kn(RZ)∼=Kn(R)⊕Kn−1(R).

Also the condition thatGis torsionfree is necessary.

IfGis finite, the conjecture above implies that the change of rings map forR→RGinduces an isomorphism

K_n(R)⊗_ZQ

∼=

−→K_n(RG)⊗_ZQ.

This is not true in general. Take for instanceR =Z,n=1 and G=Z/5. ThenK₁(Z)⊗_ZQis zero, whereas

K_n(Z[Z/5])⊗_ZQ∼=Wh(Z/5)⊗_ZQ∼=Q.

Exercise (Failure for finiteG)

Let G be a finite group G and F be a field F of characteristic zero.

Show that the map K₀(R)⊗_ZQ

∼=

−→K₀(RG)⊗_ZQis bijective if and only if G is trivial.

Lemma

Let R be a regular ring. Suppose that G is torsionfree and satisfies the K -theoretic Farrell-Jones Conjecture for torsionfree groups and regular rings. Then

Kn(RG) =0for n≤ −1;

The change of rings map K₀(R)→K₀(RG)is bijective. In particularKe₀(RG)is trivial if and only ifKe₀(R)is trivial.

Lemma

Suppose that G is torsionfree and satisfies the K -theoretic

Farrell-Jones Conjecture for torsionfree groups and regular rings. Then the Whitehead groupWh(G)is trivial.

Proof.

The idea of the proof is to study theAtiyah-Hirzebruch spectral sequenceconverging toHn(BG;K_R)whoseE²-term is given by

E_p,q² =Hp(BG,Kq(R)).

SinceRis regular by assumption, we getKq(R) =0 forq ≤ −1.

Hence the edge homomorphism yields an isomorphism K₀(R) =H₀(pt,K₀(R))−→^∼= H₀(BG;K_R)∼=K₀(RG).

Proof (continued).

We haveK₀(Z) =ZandK₁(Z) ={±1}.

We get an exact sequence

0→H₀(BG;K_Z) ={±1} →H₁(BG;K_Z)∼=K₁(ZG)

→H₁(BG;K₀(Z)) =G/[G,G]→1.

This implies

Wh(G) :=K₁(ZG)/{±g|g ∈G} ∼=0.

In particular, we get for a torsionfree groupGsatisfying theK-theoretic Farrell-Jones Conjecture for torsionfree groups and regular rings:

K_n(ZG) =0 forn≤ −1.

Ke₀(ZG) =0.

Wh(G) =0.

Every finitely dominatedCW-complexX withG=π1(X)is homotopy equivalent to a finiteCW-complex.

Every compacth-cobordismW of dimension≥6 withπ1(W)∼=G is trivial.

Exercise (Serre’sConjecture)

Suppose that the torsionfree group G satisfies the K -theoretic

Farrell-Jones Conjecture for torsionfree groups. Then G is of type FF if and only if it is of type FP.

Conjecture (L-theoretic Farrell-Jones Conjecture for torsionfree groups)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsionfree group G predicts that theassembly map

Hn(BG;L^h−∞i_R )→L^h−∞i_n (RG) is bijective for every n∈Z.

L^h−∞i_n (RG)is the algebraicL-theory ofRGwith decorationh−∞i.

L^h−∞i_R is the algebraicL-theory spectrum ofR.

Hn(pt;L^h−∞i_R )∼=πn(L^h−∞i_R )∼=L^h−∞i_n (R)forn∈Z.

IfKe_n(ZG) =0 forn≤0 and Wh(G) =0, then the decoration h−∞idoes not matter forLn(ZG).

We want to formulate a version of the Farrell-Jones Conjecture which works for all groups and rings.

This requires as input the theory ofclassifying spaces for families of subgroups.

These spaces are interesting in the own right for group theory and topology, and hence we spend the rest of this lecture about them.

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

Exercise (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 ofG-CW-complex.

The obviousG-equivariant analogs of theCellular Approximation Theoremand theWhitehead Theoremhold.

Definition (ProperG-action)

AG-spaceX is calledproperif for each pair of pointsx andy inX there are open neighborhoodsV_x ofx andW_y ofy inX such that set {g∈G|gV_x ∩W_y 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.

Exercise (Non-proper action)

Construct a closed manifold with a freeZ-action which is not proper.

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)andE G:=EVCYC(G).

Theorem (Homotopy characterization 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/ .

Exercise ((Another) Homotopy characterisation ofEF(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.

Exercise (Some models forEF(G)) Show

1 The G-spaces G/G is model for E_F(G)if and only ifF =ALL.

2 EG→BG:=G\EG is a model for theuniversal G-principal bundlefor G-principal bundles over CW -complexes.

3 A free G-CW -complex X is a model for EG if and only if X/G is 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.

Exercise (ContractibleE G/G)

Construct an infinite group G such that E G/G is contractible. Can such a group G be torsionfree?

Exercise (Maps between classifying spaces for families)

LetFandG be two families of subgroups of G. Show that the following assertions are equivalent

1 There is a G-map EF(G)→EG(G);

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

3 The projection E_F(G)×E_G(G)→E_F(G)is a G-homotopy equivalence;

4 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 realization of the full simplicial on G.

This 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 realization 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)

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.

Exercise (Rational homology of hyperbolic groups)

Let G be a hyperbolic group. Show that there exists a natural number N such that Hn(G;Q) =0holds for n≥N.

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.

LetF_n be the free group of rankn.

Denote byOut(F_n)the group of outer automorphisms.

Culler-Vogtmannhave constructed a spaceXncalledouter space on which Out(F_n)acts with finite isotropy groups. It is analogous to the Teichmüller space of a surface with the action of the mapping class group of the surface.

The spaceXncontains aspineKnwhich is an Out(Fn)-equivariant deformation retraction.

This spaceK_nis a simplicial complex of dimension(2n−3)on which the Out(Fn)-action is by simplicial automorphisms and cocompact.

Theorem (Spine of outer space)

The barycentric subdivision K_n⁰ is a finite(2n−3)-dimensional model of EOut(F_n).

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 stabilizers. 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.

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.

Exercise (Models of finite type)

Show: If there is a model for E G of finite type, then the same is true for EG and BG.

Exercise (Finitely generated homology)

Suppose that G is a hyperbolic group, a mapping class group,Out(F_n) or a cocompact discrete subgroup of a connected Lie group.

Show that then G is finitely presented and that H_i(G;Z)is finitely generated for all i ≥0.

Theorem (Discrete subgroups of Lie groups)

Let L be a Lie group with finitely many path components. Let K ⊆L be a maximal compact subgroup K . Let G⊆L be a discrete subgroup.

1 Then L/K with the left G-action is a model for E G.

2 Suppose additionally that G isvirtually torsionfree, i.e., contains a torsionfree subgroup∆⊆G of finite index.

Then we have for itsvirtual cohomological dimension vcd(G)≤dim(L/K).

Equality holds if and only if G\L is compact.

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.

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.

Proof.

ObviouslyX is aG-CW-complex with finite isotropy groups.

We have to show forH⊆Gfinite thatE G^H contractible.

We begin with the caseH 6={1}.

Because of conditions (M) and (NM) there is precisely one index i₀∈Isuch thatH is subconjugated toM_i0 and is not

subconjugated toM_i fori 6=i₀and we get a

i∈I

G/M_i

!H

= G/M_i0H

= pt.

HenceE G^H =pt.

Proof continued.

It remains to treatH ={1}. Sinceu₁is a non-equivariant homotopy equivalence andj₁is a cofibration,f₁is a

non-equivariant homotopy equivalence. HenceE Gis contractible.

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

Hp(M_i)−^∼=→Hp(G).

LetGbe one relator-group. ThenGhas a model forE Gof dimension 2, contains up to conjugacy precisely one maximal subgroupM, andMis isomorphic toZ/mfor somem≥1. Hence we get forn≥3

Hn(Z/m)−^∼=→Hn(G).

Information about E G

We will be forced when dealing with the Farrell-Jones Conjecture to deal withE Ginstead ofE G. In generalE Gis much more complicated and does not have such nice models asE G.

The following conjecture is known to be true for many groups, e.g., hyperbolic groups.

Conjecture (Finite models forE G)

There is a model of finite type for E G if and only if G is virtually cyclic.

Theorem (Lück-Weiermann)

1 mindim^G(E G)≤1+mindim^G(E G);

2 If G is virtuallyZⁿfor n ≥2, thenmindim^G(E G) =n+1and mindim^G(E G) =n;

3 There exists an extension1→Z→H →Zⁿ →1and an automorphismφof H such that the semidirect product G=HoφZsatisfies

mindim^G(E G) = n+1;

mindim^G(E G) = n+2.

In particular we get

mindim^G(E G)<mindim^G(E G).

To be continued

Stay tuned