Status and Methods of Proof (Lecture V)

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Status and Methods of Proof (Lecture V)

Wolfgang Lück Bonn Germany

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

Bonn, August 2013

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Flashback

We introduced theFarrell-Jones Conjectureand the Baum-Connes Conjecturein general

H_n^G(EVCyc(G),K_R) −^∼⁼→ H_n^G(pt,K_R) =K_n(RG);

H_n^G(EVCyc(G),L^h−∞i_R ) −^∼⁼→ H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG);

K_n^G(E G) =H_n^G(EFin(G),K^top) −^∼⁼→ H_n^G(pt,K^top) =Kn(C_r^∗(G)).

We discussed further applications of these conjectures.

Cliffhanger

Question (Status)

For which groups are the Farrell-Jones Conjecture and the Baum-Connes Conjecture known to be true?

Question (Methods of proof) What are the methods of proof?

Wolfgang Lück (HIM) Status and methods of proof Bonn, August 2013 2 / 39

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Outline

We give astatus reportof the Farrell-Jones Conjecture.

We discuss open cases and the search for potential counterexamples.

We discussmethods of proof.

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Status report of the Farrell-Jones Conjecture

There are certain generalizations of the Farrell-Jones Conjectures.

One can allowcoefficients in additive categoriesor consider fibered versionsor theversion with finite wreath products.

In what follows, theFull Farrell-Jones Conjecturewill mean the most general form with coefficients in additive categories and with finite wreath products and require it for bothK andL-theory.

The strong version encompasses twisted group ringsR_ΦG, or even crossed product ringsR∗G, and includes orientation charactersw:G→ {±1}in theL-theory setting.

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Theorem (Bartels, Echterhoff, Farrell, Lück, Reich, Roushon, Rüping, Wegner)

LetFJ be the class of groups for which the Full Farrell-Jones Conjecture holds. ThenFJ contains the following groups:

Hyperbolic groups belong toFJ; CAT(0)-groups belong toFJ;

Virtually poly-cyclic groups belong toFJ; Solvable groups belong toFJ;

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

If R is a ring whose underlying abelian group is finitely generated free, then SL_n(R)and GL_n(R)belong toFJ for all n≥2;

All arithmetic groups belong toFJ.

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Theorem (continued)

Moreover,FJ has the following inheritance properties:

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

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

If H ⊆G is a subgroup of G with[G:H]<∞and H ∈ FJ, then G∈ 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;

Many more mathematicians have made important contributions to the Farrell-Jones Conjecture, e.g.,Bökstedt, Carlsson, Davis, Ferry, Hambleton, Hsiang, Jones, Linnell, Madsen, Pedersen, Quinn, Ranicki, Rognes, Rosenthal, Tessera, Varisco, Weinberger, Yu.

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

There are manyconstructions of groups with exotic properties which arise as colimits of hyperbolic groups.

One example is the construction ofgroups with expandersdue to Gromov, seeArzhantseva-Delzant. These yieldcounterexamples to theBaum-Connes Conjecture with coefficientsdue to

Higson-Lafforgue-Skandalis.

However, our results show that these groups do satisfy the Full Farrell-Jones Conjecture and hence also the other conjectures mentioned above.

Many groups of the region ‘Hic abundant leones’in the universe of groups in the sense ofBridsondo satisfy the Full Farrell-Jones

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Davis-Januszkiewiczhave constructed exotic closed aspherical manifolds usinghyperbolization techniques. For instance there are examples which donot admit a triangulationor whose universal covering is not homeomorphic to Euclidean space.

However, in all cases the universal coverings are CAT(0)-spaces and the fundamental groups are CAT(0)-groups. Hence they satisfy the Full Farrell-Jones Conjecture and in particular the Borel Conjecture in dimension≥5.

The Baum-Connes Conjecture is open for CAT(0)-groups,

cocompact lattices in almost connected Lie groups andSL_n(Z)for n≥3, but known, for instance, for alla-T-menable groupsdue to work ofHigson-Kasparov.

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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:

Amenable groups;

Mapping class groups;

Out(Fn);

Thompson groups.

We have no good candidate for a group (or for a property of groups) for which the Farrell-Jones Conjecture may fail.

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Mini-Break

How do you feel about mathematics?

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Controlled Topology

The assembly map can be thought of anapproximationof the algebraicK- orL-theoryby a homology theory.

The basic feature between the left and right side of the assembly map is that on the left side one hasexcisionwhich is not present on the right side.

In general excision is available if one can makerepresenting cycles small.

A best illustration for this is the proof of excision for simplicial or singular homology based onsubdivisionwhose effect is to make the support of cycles arbitrary small.

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The first big step in the proof of the Farrell-Jones Conjecture is to interpret the assembly map as aforget controlmap.

Then the basic goal of the proof is obvious: Find a procedure to make the support of a representing cocycle as small as possible without changing its class, i.e.,gain control.

The following result is a prototype of this idea.

Theorem (Controlledh-Cobordism Theorem,Ferry)

Let M be a compact Riemannian manifold of dimension≥5. Then there exists an=_M >0, such that every-controlled h-cobordism over M is trivial.

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One basic idea is to pass togeometric modulesby remembering the position of a basis.

For instance, if we have a simplicial complexX, each basis element of the simplicial chain complex has a position inX, namely the barycenter of the simplex.

Similarly, one may assign to a handlebody a position in the underlying manifold.

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Given a metric spaceX, letC(X,R)be the following category:

Objects are collections{M_x}={M_x |x ∈X}, where eachMx is a finitely generated freeR-module and the support is required to be locally finite.

Morphisms{f_x,y}:{M_x} → {N_y}are given by collection of R-morphismsf_x,y:M_x →N_y respecting certain finiteness conditions so that the composition can be defined by the usual formula for the multiplication of matrices.

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IfX comes with aG-action, thenGacts onC(X;R)and we can consider theG-fixed point setC(X,R)^G. Denote byT(X;G)the full subcategory ofC(X;R)^G where we additionally require that the support of a module is cocompact.

ObviouslyT(G;R) =C(G,R)^G is the category of finitely generated freeRG-modules and hence

πn K(T(G;R))

=K_n(RG).

IfX is aG-space, then projection induces an equivalence of categoriesT(G×X;R)→ T(G;R). It induces forn∈Za homotopy equivalence after takingK-theory

πn K(T(G×X;R)) ^∼=

−→K_n(RG).

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Imposing appropriatecontrol conditionsonT(G×X;R), leads to a subcategoryT_c(G×X;R)with the property that

X 7→π∗ K(T_c(G×X;R))

yields aG-homology theory.

Theforget control map

π_n K(T_c(G×EVCyc(G);R))

→π_n K(T(G×EVCyc(G);R)) can be identified with the assembly map appearing in the K-theoretic Farrell-Jones Conjecture.

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Suppose thatG=π₁(M)for a closed Riemannian manifold with negative sectional curvature.

The idea is to use thegeodesic flowon the universal covering to gain the necessary control.

We will briefly explain this in the case, where the universal covering is the two-dimensional hyperbolic spaceH².

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Consider two points with coordinates(x₁,y₁)and(x₂,y₂)in the upper half plane model of two-dimensional hyperbolic space. We want to use the geodesic flow to make their distance smaller in a functorial fashion. This is achieved by letting these points flow towards the boundary at infinity along the geodesic given by the vertical line through these points, i.e., towards infinity in the y-direction.

There is a fundamental problem: ifx₁=x₂, then the distance between these points is unchanged. Therefore we make the following prearrangement. Suppose thaty₁<y₂. Then we first let the point(x₁,y₁)flow so that it reaches a position wherey₁=y₂. Inspecting the hyperbolic metric, one sees that the distance between the two points(x₁, τ)and(x₂, τ)goes to zero ifτ goes to infinity. This is the basic idea to gain control in the negatively curved case.

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Why is the non-positively curved case harder?

Again, consider the upper half plane, but this time equip it with the flat Riemannian metric coming from Euclidean space.

Then the same construction makes sense, but the distance between(x₁, τ)and(x₂, τ)is unchanged if we changeτ. The basic first idea is to choose afocal pointfar away, say f := (x₁+x₂)/2, τ +169356991

, and then let(x₁, τ)and(x₂, τ) flow along the rays emanating from them and passing through the focal pointf.

In the beginning the effect is indeed that the distance becomes smaller, but as soon as we have passed the focal point the distance grows again. Either one chooses the focal point very far away or uses the idea of moving the focal point towards infinity

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Coverings and Contracting Maps

Let(X,d_X)be a metric space andU an open covering of finite (topological) dimensionN. Let|U |be its nerve.

There is a canonical map

f =f^U:X → |U |, x 7→ X

U∈U

f_U(x)U, where

f_U(x) = a_U(x) P

V∈Ua_V(x);

a_U(x) = d(x,Z−U) =inf{d(x,u)|u ∈/ U}.

Suppose thatβ ≥1 is aLebesgue numberforU.

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Theorem (Contracting map)

If x,y ∈X satisfy d_X(x,y)≤ _4(N+1)^β , then we get d_{|U |}(f(x),f(y))≤ 12(N+1)²

β ·d_X(x,y).

The largerβ is, the estimate applies more often and the mapf is stronger contracting.

The largerN is, the estimate applies less often and the weakerf is contracting. IfN =∞, there is no conclusion at all.

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Axiomatic Formulation

Definition (OpenF-covering)

LetF be a family of subgroups ofGand letY be aG-space. Anopen F-coveringU is an open covering ofY satisfying

U ∈ U,g ∈G =⇒ gU ∈ U;

U ∈ U,g ∈G,gU∩U 6=∅ =⇒ gU=U;

ForU∈ U the subgroupG_U :={g∈G|gU =U}belongs toF.

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Definition (Weak Z-set condition)

A pair(X,X)satisfies theweakZ-set conditionif there exists a homotopyH:X ×[0,1]→X, such thatH₀=id_X andH_t(X)⊂X for everyt>0.

IfMis a manifold with boundary, then(M, ∂M)satisfies the weak Z-set condition because of the existence of a collar.

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Theorem (Axiomatic Formulation)

Let G be a finitely generated group. LetF be a family of subgroups of G. Suppose:

There exists a G-space X such that the underlying space X is the realization of an abstract simplicial complex;

There exists a G-space X which contains X as an open G-subspace such that the underlying space of X is compact, metrizable and contractible;

The pair(X,X)satisfies the weak Z -set condition.

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Theorem (continued)

There existswide openF-coverings, i.e.:

There is N ∈N, which only depends on the G-space X , such that for everyβ ≥1there exists an openF-coverU(β)of G×X with the following two properties:

For every g∈G and x∈X there exists U∈ U(β)such that Bβ(g)× {x} ⊂U;

The dimension of the open coverU(β)is smaller than or equal to N.

Then both the K - and L-theoretic Farrell-Jones Conjecture (with coefficients) hold for(G,F).

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An obvious choice for(X,X)isX =X =pt. But then the existence of wide open coverings impliesF =ALL.

Proof: We can chooseβ so large thatB_β(e)contains a (finite) set of generatorsS. ChooseU∈ U withB_β(e)∈U. Then we have gU∩U 6=∅and hencegU =U for allg∈S. This impliesG_U =G and henceG∈ F.

We will need the spaceX to obtain some additional spaces to maneuver open sets around in order avoid too many intersections.

The numbersN andβ conflict with each another. The larger we takeβ, the higher the chance is that many members ofU intersect.

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IfMis a closed manifold with negative sectional curvature and G=π₁(M), then the canonical choice forX isMe and forX its standard compactificationM =Me ∪∂M.e

IfGis a hyperbolic group, one uses forX theRips complexand forX =X ∪∂G, where∂Gis theboundaryof a hyperbolic group.

We consider this case in what follows.

The main technical point then is the construction of the wide open VCyc-coveringU(β).

This will be achieved with the help of a flow space FS(X). We will use a variant that is closely related to the construction

ofMineyev(2005).

Our main contribution to the flow space in the case of a hyperbolic

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Theorem (Flow space estimate)

There exists a continuous G-equivariant map j:G×X →FS(X)

such that for everyα >0there exists a numberβ =β(α)such that the following holds:

If g,h∈G with d_G(g,h)≤α and x ∈X then there isτ₀∈[−β, β]such that for allτ ∈R

d_FS(φ_τj(g,x), φ_τ+τ₀j(h,x))≤f_α(τ).

Here f_α:R→[0,∞)is a function that depends only onαand has the property thatlimτ→∞f_α(τ) =0.

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Then the next big step is to construct an appropriate openVCyc- covering on the flow space FS(X)such that the desired covering onG×X is obtained by pulling back this open covering on FS(X) withΦτ◦jfor appropriateτ.

Theorem (Long thin coverings)

There exists a natural number N such that for everyβ >0there is a VCyc-coverU ofFS(X)with the following properties:

dimU ≤N;

For every x ∈X there exists U∈ U such that

Φ_[−β,β](x) :={Φ_τ(x)|τ ∈[−β, β]} ⊆U;

G\U is finite.

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Next we explain why our strategy will not work for a smaller family thanVCyc_I.

Consider a subgroupH ⊆Gwhich can be written as an extension 1→F →H →Z→1 for a finite groupH. Chooseg ∈H which maps to a generator ofZ.

Then there arex ∈X andt ∈(0,∞)such thatφ_t(x) =gx and hx =x holds for allh∈F.

Ifαsatisfiest < α, thenΦ[−α,α](x)⊆U impliesgx ∈U and hx ∈U for allh∈F. HencegU∩U 6=∅andhU∩U 6=∅for all h∈F. This impliesg∈G_U andh∈G_U for allh∈F.

HenceG_UcontainsH.

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Cliffhanger

Question

Is the Farrell-Jones Conjecture true for all groups?

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Last but not least

Female student: “So you don’t think this is weird at all, Anatol?”

Anatol: “Absolutely, There should be anoin the second integral as opposed to an”.

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Main actor

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Status and Methods of Proof (Lecture V)