The flow space associated to a CAT(0)-space (Lecture IV)

The flow space associated to a CAT(0)-space (Lecture IV)

Wolfgang Lück Bonn Germany

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

Göttingen, June 24, 2011

Outline

We introduceCAT(0)-spacesandCAT(0)-groupsand state their main properties.

We construct theflow space FS(X)associated to a CAT(0)-space and collect its main properties.

We discuss the mainflow estimate.

CAT(0)-spaces

Definition (CAT(0)-space)

ACAT(0)-spaceorHadamard spaceis a geodesic complete metric space(X,d_X)such that any geodesic triangle∆inX satisfies the CAT(0)-inequality

d_X(x,y)≤d

R²(x,y)

for allx,y ∈∆and all comparison pointsx,y in the comparison triangle∆⊆R².

A metric spaceX is calledgeodesicif for any two pointsx,y ∈X there exists a geodesic segment joiningx andy, i.e., an isometric embeddingc: [0,d_X(x,y)]→X withc(0) =x and

c(d_X(x,y)) =y.

Ageodesic triangle∆inX consists of three pointsp,q,r and a choice of geodesic segments[p,q],[p,r]and[q,r].

Acomparison triangle∆for geodesic triangle∆is a geodesic triangle∆⊆R²given by three pointsp,q andr such that

d_X(p,q) =d_R2(p,q),d_X(p,r) =d_R2(p,r), andd_X(q,r) =d_R2(q,r).

Ifx belongs to the segment[p,q], then itscomparison pointx is the point on the geodesic[p,q]uniquely determined by

d_X(p,x) =d_R2(p,x)andd_X(x,q) =d_R2(x,q).

A simply connected complete Riemannian manifold with non-positive sectional curvature is a CAT(0)-space.

There is a unique geodesic segment joining each pairs of points and this geodesic segment various continuously with its endpoints.

IfX is a CAT(0)-space, thenX and every open ball and every closed ball inX are contractible.

Definition (Generalized geodesic)

Let(X,d_X)be a metric space. A continuous mapc:R→X is called a generalized geodesicif there arec−,c+∈R:=R`

{−∞,∞}

satisfying

c−≤c₊, c− 6=∞, c₊6=−∞,

such thatc is locally constant on the complement of the interval Ic := (c−,c+)and restricts to an isometry onIc.

Definition (Boundary of a metric space)

LetX be a metric space. Two geodesic raysc,c⁰: [0,∞)→X are calledasymptoticif there exists a constantK withd_X(c(t),c⁰(t))≤K for allt ∈[0,∞). The boundary∂X ofX is the set of asymptotic equivalence classes of rays. Denote byX =Xq∂X the disjoint union ofX and∂X.

Lemma

Let X be aCAT(0)-space and c: [0,∞)→X be a geodesic ray. Then for every x⁰ ∈X there is a unique geodesic ray c⁰: [0,∞)→X with c⁰(0) =x⁰ such that c and c⁰ are asymptotic.

In contrast to hyperbolic spaces it is in general not true for a CAT(0)- space that for two distinct elementsy,z ∈∂X there exists a geodesicc:R→X joiningy andz.

Ageneralized geodesic rayis a generalized geodesiccthat is either a constant generalized geodesic or a non-constant generalized geodesic withc−=0.

Fix a base pointx₀∈X in the CAT(0)-spaceX. For everyx ∈X, there is a unique generalized geodesic raycx such thatc(0) =x₀ andc(∞) =x. Define forr >0 the canonical projection

ρr =ρr,x₀:X →Br(x₀) byρr(x) :=cx(r).

Definition (Cone topology onX.)

LetX be a CAT(0)-space. The sets(ρr)⁻¹(V)withr >0,V an open subset ofB_r(x₀)are a basis for thecone topology onX.

The cone topology is independent of the choice of base point.

A mapf whose target isX is continuous if and only ifρr◦f is continuous for allr.

X is a compact metrizable space.

∂X ⊆X is closed andX ⊆X is dense.

The inclusionX →X is a homeomorphism onto its image which is an open subset.

IfMis a simply connected completen-dimensional Riemannian manifold with non-positive sectional curvature, then∂M isSⁿ⁻¹. There are closed topological manifoldsM constructed

byDavis-Januszkiewicz(1991)such that the universal coveringMe admits aπ1(M)-invariant CAT(0)-metric and∂Me is not

homeomorphic to a sphere andMe is not homeomorphic toRⁿ.

CAT(0)-groups

Definition (CAT(0)-group)

A (discrete) groupGis called aCAT(0)-groupif it acts properly cocompactly and isometrically on a CAT(0)-space of finite topological dimension.

A CAT(0)-groupGsatisfies:

There exists a finite modelE G.

There is a model forBGof finite type;

Gis finitely presented;

There are only finitely many conjugacy classes of finite subgroups;

Every solvable subgroup is virtuallyZⁿ;

The direct product of two CAT(0)-groups is again a CAT(0)-group;

Limit groups in the sense of Sela are CAT(0)-groups;

Coxeter groups are CAT(0)-groups;

The word-problem and the conjugation-problem are solvable.

Question

Is every hyperbolic group aCAT(0)-group?

The flow space of a metric space

Throughout this section let(X,d_X)be a metric space.

Definition (Flow space)

LetFS=FS(X)be the set of all generalized geodesics inX; We define ametricon FS(X)by

d_FS(X)(c,d):=

Z

R

d_X(c(t),d(t)) 2e^|t| dt.

Define aflow

Φ: FS(X)×R→FS(X)

byΦ_τ(c)(t) =c(t+τ)forτ ∈R,c ∈FS(X)andt∈R.

Lemma

The mapΦis a continuous flow and we have for c,d ∈FS(X)and τ, σ∈R

d_FS(X) Φτ(c),Φσ(d)

≤ e^|τ|·d_FS(X)(c,d) +|σ−τ|.

Proof:

We estimate forc ∈FS(X)andτ ∈R: d_FS(X) c,Φ_τ(c)

= Z

R

d_X c(t),c(t+τ) 2e^|t| dt

≤ Z

R

|τ| 2e^|t| dt

= |τ| · Z

R

1 2e^|t| dt

= |τ|.

We estimate forc,d ∈FS(X)andτ ∈R

d_FS(X) Φ_τ(c),Φ_τ(d)

= Z

R

d_X c(t+τ),d(t+τ) 2e^|t| dt

= Z

R

d_X c(t),d(t) 2e^|t−τ| dt

≤ Z

R

d_X c(t),d(t) 2e^|t|−|τ| dt

= e^|τ|· Z

R

d_X c(t),d(t) 2e^|t| dt

= e^|τ|·d_FS(X)(c,d).

The two inequalities above together with the triangle inequality imply forc,d ∈FS(X)andτ, σ∈R

d_FS(X) Φ_τ(c),Φ_σ(d)

= d_FS(X) Φτ(c),Φσ−τ◦Φτ(d)

≤ d_FS(X) Φτ(c),Φτ(d)

+d_FS(X) Φτ(d),Φσ−τ ◦Φτ(d)

≤ e^|τ|·d_FS(X)(c,d) +|σ−τ|.

Lemma

Let c,d:R→X be generalized geodesics. Consider t₀∈R. d_X c(t₀),d(t₀)

≤ e^|t0|·d_FS(c,d) +2;

If d_FS(c,d)≤2e^−|t0|−1, then d_X c(t₀),d(t₀)

≤ p

4e^|t0|+1·p

d_FS(c,d).

In particular, c7→c(t₀)defines a uniform continuous map FS(X)→X .

Proof of the first assertion

We abbreviateD:=d_X(c(t₀),d(t₀)).

We get

d_X(c(t),d(t))≥D−d_X c(t₀),c(t)

−d_X d(t₀),d(t)

≥D−2·|t−t₀|.

d_FS(X)(c,d) =

Z +∞

−∞

d_X(c(t),d(t)) 2e^|t| dt

≥

Z D/2+t₀

−D/2+t₀

D−2· |t−t₀| 2e^|t| dt

=

Z D/2

−D/2

D−2· |t|

2e^|t+t0| dt

≥

Z D/2

−D/2

D−2· |t|

2e^|t|+|t0| dt

= e^−|t0|· Z D/2

−D/2

D−2· |t|

2e^|t| dt

= e^−|t0|·

2·e^−D/2+D−2

≥ e^−|t0|·(D−2).

Lemma The maps

FS(X)−FS(X)^R → R, c 7→c−; FS(X)−FS(X)^R → R, c 7→c₊, are continuous.

Lemma

Let(c_n)n∈Nbe a sequence inFS(X). Then it converges uniformly on compact subsets to c∈FS(X)if and only if it converges to c with respect to d_FS(X).

Lemma

The flow spaceFS(X)is sequentially closed in the space of all maps R→X with respect to the topology of uniform convergence on compact subsets.

Definition (Proper metric space)

A metric space is calledproper if every closed ball is compact.

Lemma

If(X,d_X)is a proper metric space, then(FS(X),d_FS(X))is a proper metric space.

Proof:

LetR >0 andc ∈FS(X).

It suffices to show that the closed ballB_R(c)in FS(X)is sequentially compact.

Let(cn)n∈Nbe a sequence inB_R(c). There isR⁰ >0 such that c_n(0)∈B_R⁰(c(0)). By assumptionB_R⁰(c(0))is compact.

Now we can apply theArzelà-Ascoli Theorem.

Thus after passing to a subsequence there isd:R→X such that c_n→d uniformly on compact subsets.

Lemma

Let(X,d_X)be a proper metric space and t₀∈R. Then the evaluation mapFS(X)→X defined by c 7→c(t₀)is uniformly continuous and proper.

Proof:

We have already shown that the map is uniformly continuous To show that is is also proper, it suffices to show that preimages of closed balls have finite diameter.

Ifd_X(c(t₀),d(t₀))≤r, thend_X(c(t),d(t))≤r +2|t−t₀|. Thus d_FS(c,d)≤

Z

R

r +2|t−t₀| 2e^|t| dt, providedd_X c(t₀),d(t₀)

≤r.

Lemma

Let G act isometrically, properly and cocompactly on the proper metric space(X,d_X). Then action of G on(FS(X),d_FS)is also isometric, proper and cocompact.

Proof:

The action ofGon FS(X)is isometric.

The map FS(X)→X defined byc 7→c(0)isG-equivariant, continuous and proper.

The existence of such a map implies that theG-action on FS(X)is also proper and cocompact.

Lemma

The subspaceFS(X)^Ris closed inFS(X).

LetX be a metric space. Forc∈FS(X)andT ∈[0,∞], define c|_[−T,T]∈FS(X)by

c|_[−T,T](t) :=







c(−T) ift≤ −T; c(t) if −T ≤t≤T; c(T) ift≥T.

We denote by FS(X)_f :=n

c ∈FS(X)−FS(X)^R

c− >−∞,c₊ <∞o

∪FS(X)^R the subspace of finite geodesics.

Lemma The map

H: FS(X)×[0,1]→FS(X)

defined by H_τ(c) :=c|_{[ln(τ),−ln(τ)]}is continuous and satisfies H₀=id_FS(X)and H_τ(c)∈FS(X)_f forτ >0.

The flow space of a CAT(0)-space

Example (Flow space of a manifold of non-positive sectional curvature)

LetM be a simply connected complete Riemannian manifold of non-positive sectional curvature.

Recall thatMis a CAT(0)-space.

Put

P := {(a−,a₊)∈R×R|a− <∞,a₊>−∞,a− ≤a₊};

∆ = {(a,a)∈R×R| −∞<a<∞}.

Define maps

f:STM×P → FS(M), (v,a−,a+)7→c(v)_[a−,a+]; p:STM×∆ → M, (v,a)7→c_v(a),

Example (continued)

The mapf is compatible with the obvious flows.

Then we obtain pushout

STM×∆ ^{i //}

p

STM×P f

M ^{j //}FS(M)

wherei is the inclusion andj:M →FS(X)sendsx to const_x. In particularf induces a homeomorphism

STM×(P−∆)−^∼=→FS(M)−FS(M)^R.

Definition (End points of a geodesic) Forc ∈FS(X)we definec(∞)∈X by

c(∞) := lim

t→∞c(t) =

(c(c+) ifc+<∞;

[c|_[0,∞)] ifc₊=∞.

Definec(−∞)analogously.

Lemma The maps

FS(X)−FS(X)^R → X, c7→c(−∞);

FS(X)−FS(X)^R → X, c7→c(∞), are continuous.

The two maps appearing above cannot be continuosly extended to FS(X)by the following observation.

Letc be a generalized geodesic withc₊<∞andc− =∞. Then c(−∞) 6= c(∞);

d_FS(c,const_c(∞)) ≤ e^c+/2;

τ→∞lim Φ_τ(c) = const_c(∞);

τ→∞lim Φτ(c)

(−∞) = c(∞);

Φ_τ(c)(−∞) = c(−∞) for allτ >0;

τ→∞lim (Φ_τ(c)(−∞)) = c(−∞).

Theorem (Embedding the flow space) If X is proper as a metric space, then the map

E: FS(X)−FS(X)^R→R×X ×X ×X ×R defined by E(c) := (c−,c(−∞),c(0),c(∞),c₊)is injective and continuous. It is a homeomorphism onto its image.

Lemma

If X is proper as a metric space and its covering dimensiondimX is

≤N, thendimX ≤N.

Proof:

LetU ={U_i |i ∈I}be an open covering ofX.

For everyx ∈X there arer_x,W_x ⊆B_rx(x₀)andU_x ∈ U such that x ∈ρ⁻¹_rx (Wx)⊂Ux.

SinceX is compact, a finite number of the setsρ⁻¹_rx (W_x)coverX. Note thatρr =ρr|_B

r0(x0)◦ρr⁰ and hence ρ⁻¹_r (W) =ρ⁻¹_r0 (ρr|_B

r0(x0)(W))ifr⁰ >r.

Therefore we can refineU to a finite coverV, such that there isr and a finite coverW ofB_r(x₀)such that

V =ρ⁻¹_r (W) :={ρ⁻¹_r (W)|W ∈ W}.

Wolfgang Lück (Bonn, Germany) The flow space Göttingen, June 24, 2011 27 / 35

Theorem (Dimension of the flow space)

Assume that X is proper and thatdimX ≤N. Then dim FS(X)−FS(X)^R

≤3N+2.

Proof:

Every compact subsetK of FS(X)−FS(X)^Ris homeomorphic to a compact subset ofR×X×X×X×R.

Hence its topological dimension satisfies dim(K)≤dim R×X ×X ×X ×R

=2 dim(R) +2 dim(X) +dim(X)≤3N+2.

One shows that FS(X)−FS(X)^Rhas a countable basis for its topology.

Now dim FS(X)−FS(X)^R

≤3N+2 follows from standard result of dimension theory.

The homotopy action on B

(x

)

TheG-action onX induces anG-action onX.

For technical reasons we will not take the spaceX as the space appearing in the axiomatic approach as we have done it for hyperbolic groups. We will take the closed ballB_R(x₀)for some base pointx₀and some very large real numberR.

The prize to pay is that we do not obtain aG-action onB_R(x₀)but at least the following homotopyG-action.

Definition (The homotopyG-action onBR(x0))

Define ahomotopyG-action(ϕ^R,H^R)onB_R(x)as follows.

Forg∈G, we define the map

ϕ^R_g:B_R(x₀)→B_R(x₀) byϕ^R_g(x) :=ρ_R,x0(gx).

Forg,h∈Gwe define the homotopy H_g,h^R :ϕ^R_g ◦ϕ^R_h 'ϕ^R_gh

byH_g,h^R (x,t) :=ρ_R,x0 t·(ghx) + (1−t)·(g·ρ_R,x0(hx)) .

x₀

x

gx B_R(x₀)

ϕ^R_g(x)

x₀

x

hx B_R(x₀)

y :=ρR,x₀(hx)

gy ghx H_g,h^R (x,t)

It turns out that the more obvious homotopy given by convex combination(x,t)7→t·ϕ^R_gh(x) + (1−t)·ϕ^R_g ◦ϕ^R_h(x)is not appropriate for our purposes.

Notice thatH_g,h^R is indeed a homotopy fromϕ^R_g ◦ϕ^R_h toϕ_gh since H_g,h^R (x,0) = ρ_R,x0 0·(ghx) +1·(g·ρ_R,x0(hx))

= ρ_R,x0 g·ρ_R,x0(hx)

= ϕ^R_g ◦ϕ^R_h(x),

and

H_g,h^R (x,1) = ρ_R,x0 1·(ghx) +0·(g·ρ_R,x0(hx)

= ρ_R,x0(ghx)

= ϕ^R_gh(x).

The map ι

Definition (The mapι) Define the map

ι:G×X →FS(X)

by sending(g,x)∈G×X to the generalized geodesiccgx₀,gx fromgx₀ togx.

The flow estimate

Theorem (The flow estimate)

Letβ,L>0. For allδ >0there are T,r >0with the following property:

For x₁,x₂∈X with d_X(x₁,x₂)≤β, x ∈B_r+L(x₁)there isτ ∈[−β, β]

such that

d_FS Φ_T(c_x1,ρr,x

1(x)),Φ_T+τ(c_x2,ρr,x

2(x))

≤ δ.

x₁

x Br(x₁)

B_r(x₂)