On the Farrell-Jones Conjecture and its applications

On the Farrell-Jones Conjecture and its applications

Wolfgang Lück Bonn Germany

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

November 2010, Göttingen

Outline

Motivation and statement of the Farrell-Jones Conjecture Some prominent conjectures

The status of the Farrell-Jones Conjecture Open problems

We have prepared more slides than we will probably show.

Algebraic K -and L-theory

Definition (Projective class groupK0(R)) Define theprojective class groupof a ringR

K₀(R)

to be the abelian group defined by:

Generators: Isomorphism classes[P]of finitely generated projective R-modulesP.

Relations: We get[P₀] + [P₂] = [P₁]for every exact sequence 0→P₀→P₁→P₂→0 of finitely generated projectiveR-modules.

Thereduced projective class groupKe₀(R)is the quotient ofK₀(R) by the subgroup generated by the classes of finitely generated freeR-modules.

LetP be a finitely generated projectiveR-module. It isstably free, i.e.,P⊕R^m ∼=Rⁿfor appropriatem,n∈Z, if and only if[P] =0 in Ke₀(R).

Definition (K1-groupK1(R)) Define theK₁-groupof a ringR

K₁(R):=GL(R)/[GL(R),GL(R)].

An invertible matrixA∈GL(R)can be reduced byelementary row and column operationsand(de-)stabilizationto the trivial empty matrix if and only if[A] =0 holds in thereduced K₁-group

Ke₁(R):=K₁(R)/{±1}.

Definition (Whitehead group)

TheWhitehead groupof a groupGis defined to be Wh(G)=K₁(ZG)/{±g|g ∈G}.

Bass (1968)andQuillen (1973)have definedK_n(R)for alln∈Z. K-theory has some basic feature such ascompatibility with products,Morita equivalence,Bass-Heller-Swan decomposition andlocalization sequences.

L-groupsLn(R)are defined in a similar way but now for quadratic forms.

In contrast toK-theory theL-groups arefour-periodic, i.e., L_n(R) =L_n+4(R)

In general algebraicK-andL-theory arevery hard to computebut ofhigh significance.

Motivation and statement of the Farrell-Jones Conjecture

LetH∗ be a (generalized) homology theory. It satisfies:

Suspension

H_n(BZ) =H_n(S¹)∼=H_n(pt)⊕H_n−1(pt) =H_n(B{1})⊕H_n−1(B{1}).

Mayer-Vietoris-sequence IfG=G₁∗_G

0G₂, then we get a long exact sequence

· · · → H_n(BG₀)→ H_n(BG₁)⊕ H_n(BG₂)→ H_n(BG)

→ H_n−1(BG₀)→ H_n−1(BG₁)⊕ H_n−1(BG₂)→ · · ·

LetR be a regular ring. Then:

Bass-Heller-Swan (1964)have shown:

K_n(R[Z])∼=K_n(R)⊕K_n−1(R) =K_n(R[{1}])⊕K_n−1(R[{1}]).

IfG=G₁∗_G

0G₂andG₀,G₁andG₂are torsionfree and belong to a certain class CL, thenWaldhausen (1978)has established the exact sequence

· · · →K_n(R[G₀])→K_n(R[G₁])⊕K_n(R[G₂])→K_n(R[G])

→Kn−1(R[G₀])→Kn−1(R[G₁])⊕Kn−1(R[G₂])→ · · ·

This raises the question: Is there a generalized homology theory H_∗ satisfying

H_n(BG)∼=K_n(RG)

for all torsionfree groupsGandn∈Z, whereR is a fixed regular ring?

If yes, we must have for alln∈Z

H_n(pt) =K_n(R).

Hence our candidate forH_∗isH∗(−;K_R), the generalized homology theory associated to the (non-connective)K-theory spectrumK_R of the ringR.

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

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

Hn(BG;K_R)→Kn(RG) is bijective for all n∈Z.

The version of the Farrell-Jones Conjecture above is not true for finite groups unless the group is trivial.

For instance we get for a finite groupGandR =C: K₀(CG) = R_C(G);

H₀(BG;K_C)⊗_ZQ = Q.

Also the condition regular is needed in general.

Namely, we have

Kn(R[Z]) = Kn(R)⊕Kn−1(R)⊕NKn(R)⊕NKn(R);

H_n(BZ;K_R) = K_n(R)⊕K_n−1(R).

Conjecture (Farrell-Jones Conjecture)

The Farrell-Jones Conjecturefor K -theory or L-theory respectively with coefficients in R predicts that the assembly map

H_n^G(EVCyc(G),K_R)→H_n^G(pt,K_R) =Kn(RG) or

H_n^G(EVCyc(G),L^h−∞i_R )→H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG) respectively is bijective for all n∈Z.

There is a more complicated version of the Farrell-Jones Conjecture which may be true for all groupsGand ringsRand makes also sense for twisted group rings and allows orientation homomorphisms inL-theory.

In the sequel we will refer to this general version.

Conjecture (Baum-Connes Conjecture)

The Baum-Connes Conjecturepredicts that the assembly map K_n^G(E_Fin(G))→K_n(C_r^∗(G))

is bijective for all n∈Z.

Conjecture (Bost Conjecture)

The Bost Conjecturepredicts that the assembly map K_n^G(EFin(G))→Kn(l¹(G)) is bijective for all n∈Z.

Some prominent conjectures

Construction of idempotents inRG

Suppose thatg ∈Ghas finite order|g|. PutN=P|g|

i=1gⁱ. Then N·N =|g| ·N.

If|g|is invertible inRand different from 1, thenRGcontains a non-trivial idempotent, namely _|g|^N.

Conjecture (Kaplansky Conjecture)

TheKaplansky Conjecturesays for a torsionfree group G and a field F that0and1are the only idempotents in FG.

Conjecture (Vanishing ofKe0(ZG)for torsionfreeG) If G is torsionfree, then

Ke₀(ZG) ={0}.

Conjecture (Vanishing of Wh(G)for torsionfreeG) If G is torsionfree, then

Wh(G) ={0}.

Conjecture (Novikov Conjecture)

TheNovikov Conjecture for Gpredicts for a closed oriented manifold M together with a map f:M→BG that for any x ∈H^∗(BG)thehigher signature

sign_x(M,f):=hL(M)∪f^∗x,[M]i is an oriented homotopy invariant of(M,f).

Iff:M →Nis a homotopy equivalence of closed aspherical manifolds, then the Novikov Conjecture predicts

f∗L(M) =L(N).

Conjecture (Borel Conjecture)

TheBorel Conjecture for Gpredicts for two closed aspherical manifolds M and N withπ1(M)∼=π1(N)∼=G that any homotopy equivalence M →N is homotopic to a homeomorphism and in particular that M and N are homeomorphic.

The Borel Conjecture can be viewed as the topological version of Mostow rigidity. A special case of Mostow rigidity says that any homotopy equivalence between closed hyperbolic manifolds of dimension≥3 is homotopic to an isometric diffeomorphism.

The Borel Conjecture is not true in the smooth category by results ofWallandFarrell-Jones.

Conjecture (Gromov)

If G is a torsionfree hyperbolic group whose boundary is a standard sphere, then there is a closed aspherical manifold M with G=π₁(M).

There are further interesting prominent conjectures byBassand byMoodyand conjectures aboutL²-invariantsand aboutPoincaré duality groups, which we do not state.

One of the basic features of the Farrell-Jones Conjecture is that it implies all the conjectures mentioned above, where in some cases one has to assume dim≥5.

The Farrell-Jones Conjecture is the basic ingredient for computations ofKn(RG)andLn(RG).

Status of the Farrell-Jones Conjectures

Theorem (Main Theorem

(Bartels-Echterhoff-Farrell-Lück-Wegner Reich (2008-2012)) LetFJ be the class of groups for which both the K -theoretic and the L-theoretic Farrell-Jones Conjectures (in its most general form) holds.

It has the following properties:

Hyperbolic groups belong toFJ;

If G₁and G₂belong toFJ, then G₁×G₂and G₁∗G₂belong to FJ;

Let{G_i |i ∈I}be a directed system of groups (with not

necessarily injective structure maps) such that G_i ∈ FJ for i ∈I.

Thencolimi∈IG_i belongs toFJ;

If H is a subgroup of G and G∈ FJ, then H ∈ FJ;

Theorem (continued)

CAT(0)-groups belong toFJ;

Virtually poly-cyclic groups belong toFJ;

Cocompact lattices in almost connected Lie groups belong toFJ; All3-manifold groups belong toFJ.

The groups above are certainly wild inBridson’suniverse of groups.

Examples are:

Lacunary hyperbolic groupsin the sense ofOlshanskii-Osin-Sapir (2009);

Tarski monsters, i.e., groups which are not virtually cyclic and whose proper subgroups are ofm orderpfor a fiyed orimep;

Limit groupsin the sense ofZelaare CAT(0)-groups (Alibegovic-Bestvina (2006)).

Arzhantseva-Delzant (2008 )have constructed groups with expanders as colimits of hyperbolic groups.

These yieldcounterexamplesto theBaum-Connes Conjecture with coefficientsdue toHigson-Lafforgue-Skandalis (2002).

However, our results show that these groups do satisfy the

Farrell-Jones Conjecture, the Bost Conjecture and hence also the other conjectures such as the conjectures due to Borel and Novikov mentioned above.

In particular the results ofHigson-Lafforgue-Skandalis (2002) show that the mapKn(Aol¹G)→Kn(AoC^∗_r G)is not bijective.

Davis- Januszkiewics (1991)have constructed exotic closed aspherical manifolds usinghyperbolization techniques. For instance there are examples which donot admit a triangulationor whoseuniversal covering is not homeomorphic to Euclidean space.

However, in all cases the universal coverings are CAT(0)-spaces and hence the fundamental groups are CAT(0)-groups.

Hence by our main theorem they satisfy the Farrell-Jones Conjecture and hence the Borel Conjecture in dimension≥5.

Open problems

What are candidates for groups or closed aspherical manifolds for which the conjectures due to Farrell-Jones, Novikov or Borel may be false?

There are still many interesting groups for which the Farrell-Jones Conjecture is open.

Examples are:

Solvable groups;

Amenable groups;

Sln(Z)forn≥3;

Mapping class groups;

Out(Fn);

Thompson groups.

Can methods of proof for the Baum-Connes or the Farrell-Jones Conjecture be transferred from one setting to the other?

Comparing the Farrell-Jones and the Baum-Connes Conjecture

H_n^G(EFin(G);L^p

Z)[1/2] ^{∼= //}

∼=

L^p_n(ZG)[1/2]

∼=

H_n^G(E_Fin(G);L^p

R)[1/2] ^{∼= //}

∼=

L^p_n(RG)[1/2]

∼=

H_n^G(E_Fin(G);L^p_C∗

r(?;R))[1/2] ^∼= //

∼=

L^p_n(C_r^∗(G;R))[1/2]

∼=

H_n^G(E_Fin(G);K^top

R )[1/2] ^{∼= //}

Kn(C_r^∗(G;R))[1/2]

H_n^G(E_Fin(G);K^top

C )[1/2] ^{∼= //}K_n(C_r^∗(G;C))[1/2]

Examples of Computations

Theorem (Torsionfree hyperbolic groups)

If G is a torsionfree hyperbolic group, then we get isomorphisms

H_n(BG;K_R)⊕









M

(C),C⊆G,C6=1 C maximal cyclic

NK_n(R)⊕NK_n(R)









∼=

−→ K_n(RG),

and

H_n(BG;L^h−∞i_R ) −^∼=→ L^h−∞i_n (RG).

The Baum-Connes Conjecture and the Bost Conjecture are also known to be true for hyperbolic groups and reduce therefore for obvious reasons for a torsionfree hyperbolic group to

Comparing algebraic and topological K -theory

Theorem (L. (2002))

Let G be a group. Let T be the set of conjugacy classes(g)of elements g ∈G of finite order. There is a commutative diagram

L

p+q=n

L

(g)∈TH_p(BC_Ghgi;C)⊗_ZK_q(C) //

K_n(CG)⊗_ZC

L

p+q=n

L

(g)∈TH_p(BC_Ghgi;C)⊗_ZK_q^top(C) ^//K_n^top(C_r^∗(G))⊗_ZC

About the proof

One needs to interpret the assembly map which is easiest described in terms of homotopy theory as aforget control homomorphism.

Then the task is to show how toget control.

This is achieved for hyperbolic and CAT(0)-groups by constructing flow spaceswhich mimic the geodesic flow on a Riemannian manifold with negative or non-positive sectional curvature.

The proof of the inheritance results is ofhomotopy theoretic nature.

Poly-cyclic groups are handled bytransfer methods.