Contracting Boundaries of CAT(0) Spaces Ruth Charney Dubrovnik, - - PowerPoint PPT Presentation

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Contracting Boundaries of CAT(0) Spaces Ruth Charney Dubrovnik, July 2011 Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 1 / 18 Motivation X = complete hyperbolic metric space. Visual boundary of X : X = {

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Contracting Boundaries of CAT(0) Spaces

Ruth Charney Dubrovnik, July 2011

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 1 / 18

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Motivation

X = complete hyperbolic metric space. Visual boundary of X: ∂X = {geodesic rays α : [0, ∞) → X}/ ∼ where α ∼ β if they have bounded Hausdorff distance. Topology on ∂X: N(α, r, ǫ) = {β | d(α(t), β(t)) < ǫ, 0 ≤ t < r}

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 2 / 18

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Properties of ∂X, X hyperbolic If X is proper, then X ∪ ∂X is compact. Quasi-isometries f : X → Y induce homeomorphisms ∂f : ∂X → ∂Y . In particular, ∂G is well-defined for a hyperbolic group G. ∂X is a visibility space, i.e. for any two points x, y ∈ ∂X, ∃ a geodesic γ with γ(∞) = x and γ(−∞) = y. Nice dynamics: hyperbolic isometries g ∈ Isom(X) act on ∂X with “north-south dynamics.”

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 3 / 18

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Now suppose X is a complete CAT(0) space. Can define ∂X in the same way, but properties are not as nice.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 4 / 18

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Now suppose X is a complete CAT(0) space. Can define ∂X in the same way, but properties are not as nice. If X is proper, then X ∪ ∂X is compact. Quasi-isometries f : X → Y do NOT necessarily induce homeomorphisms ∂f : ∂X → ∂Y , so ∂G is not well-defined for a CAT(0) group G (Croke-Kleiner). ∂X is a NOT a visibility space (eg. X = R2). Dynamics of g ∈ Isom(X) acting on ∂X ???

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 4 / 18

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Now suppose X is a complete CAT(0) space. Can define ∂X in the same way, but properties are not as nice. If X is proper, then X ∪ ∂X is compact. Quasi-isometries f : X → Y do NOT necessarily induce homeomorphisms ∂f : ∂X → ∂Y , so ∂G is not well-defined for a CAT(0) group G (Croke-Kleiner). ∂X is a NOT a visibility space (eg. X = R2). Dynamics of g ∈ Isom(X) acting on ∂X ??? Certain isometries of a CAT(0) space X behave nicely. These are known as rank one isometries.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 4 / 18

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Rank one isometries

Definition (Ballmann-Brin)

A geodesic α is rank one if it does not bound a half-flat. An isometry g ∈ Isom(X) is rank one if it has a rank one axis.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 5 / 18

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Rank one isometries

Definition (Ballmann-Brin)

A geodesic α is rank one if it does not bound a half-flat. An isometry g ∈ Isom(X) is rank one if it has a rank one axis. Ballmann-Brin-Eberlein, Schroeder-Buyalo, Kapovich-Leeb, Drutu-Moses-Sapir, Bestvina-Fujiwara, Hamenstadt, Sageev-Caprace,. . .

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 5 / 18

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Rank one isometries

Definition (Ballmann-Brin)

A geodesic α is rank one if it does not bound a half-flat. An isometry g ∈ Isom(X) is rank one if it has a rank one axis. Ballmann-Brin-Eberlein, Schroeder-Buyalo, Kapovich-Leeb, Drutu-Moses-Sapir, Bestvina-Fujiwara, Hamenstadt, Sageev-Caprace,. . . General philosophy: Rank one isometries of a CAT(0) space behave nicely because their axes behave like geodesics in a hyperbolic space.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 5 / 18

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Definition (Bestvina-Fujiwara)

A geodesic α is D-contracting if for any ball B disjoint from α, the projection of B on α has diameter at most D. A geodesic is contracting if it is D-contracting for some D.

< D

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 6 / 18

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Definition (Bestvina-Fujiwara)

A geodesic α is D-contracting if for any ball B disjoint from α, the projection of B on α has diameter at most D. A geodesic is contracting if it is D-contracting for some D.

< D

Contracting geodesics satisfy a thin triangle property.

! x " y z (D) ! thin

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 6 / 18

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Clearly, α contracting ⇒ α is rank one.

Theorem (B-F)

If X proper CAT(0) space and α is periodic, then α is rank one ⇔ it is contracting.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 7 / 18

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Clearly, α contracting ⇒ α is rank one.

Theorem (B-F)

If X proper CAT(0) space and α is periodic, then α is rank one ⇔ it is contracting. For non-periodic geodesics, α rank one α contracting.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 7 / 18

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Clearly, α contracting ⇒ α is rank one.

Theorem (B-F)

If X proper CAT(0) space and α is periodic, then α is rank one ⇔ it is contracting. For non-periodic geodesics, α rank one α contracting. Examples:

2

H ! !

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 7 / 18

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Clearly, α contracting ⇒ α is rank one.

Theorem (B-F)

If X proper CAT(0) space and α is periodic, then α is rank one ⇔ it is contracting. For non-periodic geodesics, α rank one α contracting. Examples:

2

H ! !

!

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 7 / 18

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Contracting Boundary

Back to boundaries:

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 8 / 18

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Contracting Boundary

Back to boundaries: Consider the subspace of ∂X consisting of all contracting rays. Define the contracting boundary of X ∂cX = {contracting rays α : [0, ∞) → X}/ ∼ with the subspace topology ∂cX ⊂ ∂X.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 8 / 18

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Contracting Boundary

Back to boundaries: Consider the subspace of ∂X consisting of all contracting rays. Define the contracting boundary of X ∂cX = {contracting rays α : [0, ∞) → X}/ ∼ with the subspace topology ∂cX ⊂ ∂X.

Examples

(1) If X is hyperbolic, then ∂cX = ∂X.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 8 / 18

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Contracting Boundary

Back to boundaries: Consider the subspace of ∂X consisting of all contracting rays. Define the contracting boundary of X ∂cX = {contracting rays α : [0, ∞) → X}/ ∼ with the subspace topology ∂cX ⊂ ∂X.

Examples

(1) If X is hyperbolic, then ∂cX = ∂X. (2) X = first example above, the ∂cX = ∂H2\{pt} ∼ = (0, 1).

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 8 / 18

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Contracting Boundary

Back to boundaries: Consider the subspace of ∂X consisting of all contracting rays. Define the contracting boundary of X ∂cX = {contracting rays α : [0, ∞) → X}/ ∼ with the subspace topology ∂cX ⊂ ∂X.

Examples

(1) If X is hyperbolic, then ∂cX = ∂X. (2) X = first example above, the ∂cX = ∂H2\{pt} ∼ = (0, 1). (3) If X = X1 × X2, then ∂cX = ∅

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 8 / 18

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This subspace, ∂cX, should behave like a hyperbolic boundary. Properties of ∂X, for X hyperbolic: If X is proper, then X ∪ ∂X is compact. Quasi-isometries f : X → Y induce homeomorphisms ∂f : ∂X → ∂Y . In particular, ∂G is well-defined for a hyperbolic group G. ∂X is a visibility space, i.e. for any two points x, y ∈ ∂X, ∃ a geodesic γ with γ(∞) = x and γ(−∞) = y. hyperbolic isometries g ∈ Isom(X) act on ∂X with “north-south dynamics.” Q: Are the analogous true for ∂cX of a CAT(0) space?

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 9 / 18

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Properties of ∂cX

Theorem

Suppose X is proper. The subspace of D-contracting rays is compact, hence ∂cX is σ-compact (a countable union of compact subspaces). Proof: Follows easily from lemmas in Bestvina-Fujiwara.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 10 / 18

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Properties of ∂cX

Theorem

Suppose X is proper. The subspace of D-contracting rays is compact, hence ∂cX is σ-compact (a countable union of compact subspaces). Proof: Follows easily from lemmas in Bestvina-Fujiwara.

Theorem

Let x ∈ ∂cX and y ∈ ∂X, then there exists a geodesic γ in X such that γ(∞) = x and γ(−∞) = y. In particular, ∂cX is a visibility space.

x y

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 10 / 18

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Properties of ∂cX

Theorem

Suppose X is proper. The subspace of D-contracting rays is compact, hence ∂cX is σ-compact (a countable union of compact subspaces). Proof: Follows easily from lemmas in Bestvina-Fujiwara.

Theorem

Let x ∈ ∂cX and y ∈ ∂X, then there exists a geodesic γ in X such that γ(∞) = x and γ(−∞) = y. In particular, ∂cX is a visibility space.

x y

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 11 / 18

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Main Theorem

Theorem

A quasi-isometry of CAT(0) spaces f : X → Y induces a homeomorphism ∂f : ∂cX → ∂cY . In particular, ∂cG is well-defined for a CAT(0) group G.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 12 / 18

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Main Theorem

Theorem

A quasi-isometry of CAT(0) spaces f : X → Y induces a homeomorphism ∂f : ∂cX → ∂cY . In particular, ∂cG is well-defined for a CAT(0) group G. Idea of proof. Recall, a ray α is D-contracting if for any ball B disjoint from α, the projection of B on α has diameter at most D. Problem: projection does not behave nicely under quasi-isometry. Need a characterization of contracting ray which does.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 12 / 18

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Divergence: For α a (bi-infinte) geodesic, divα(r) = inf{ℓ(p) | p a path in X\B(r, α(0)) from α(−r) to α(r)}

!(0)

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 13 / 18

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Divergence: For α a (bi-infinte) geodesic, divα(r) = inf{ℓ(p) | p a path in X\B(r, α(0)) from α(−r) to α(r)}

!(0)

Lower divergence: For α a geodesic ray, define divα(r) = inf{ℓ(p) | p a path in X\B(r, α(t)) from α(t − r) to α(t + r), t ∈ [r, ∞)}

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 13 / 18

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Divergence: For α a (bi-infinte) geodesic, divα(r) = inf{ℓ(p) | p a path in X\B(r, α(0)) from α(−r) to α(r)}

!(0)

Lower divergence: For α a geodesic ray, define divα(r) = inf{ℓ(p) | p a path in X\B(r, α(t)) from α(t − r) to α(t + r), t ∈ [r, ∞)}

Remark

These are different even for a bi-infinite geodesics. Eg, if X = R2 ∨ R2 and α passes through 0, then divα(r) = ∞, while divα(r) = πr.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 13 / 18

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Lemma

For a ray α in X, TFAE

1 divα is super-linear. 2 divα is at least quadratic. 3 α is contracting. Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 14 / 18

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Lemma

For a ray α in X, TFAE

1 divα is super-linear. 2 divα is at least quadratic. 3 α is contracting.

Theorem

A quasi-isometry of CAT(0) spaces f : X → Y induces a homeomorphism ∂f : ∂cX → ∂cY . In particular, ∂cG is well-defined for a CAT(0) group G.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 14 / 18

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Theorem

A quasi-isometry of CAT(0) spaces f : X → Y induces a homeomorphism ∂f : ∂cX → ∂cY . In particular, ∂cG is well-defined for a CAT(0) group G.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 15 / 18

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Theorem

A quasi-isometry of CAT(0) spaces f : X → Y induces a homeomorphism ∂f : ∂cX → ∂cY . In particular, ∂cG is well-defined for a CAT(0) group G. Proof: Let α be a contracting ray in X. Step 1: Show f (α) stays bounded distance from some geodesic ray β in Y .

x

R quadratic in R !

y Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 15 / 18

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Theorem

A quasi-isometry of CAT(0) spaces f : X → Y induces a homeomorphism ∂f : ∂cX → ∂cY . In particular, ∂cG is well-defined for a CAT(0) group G. Proof: Let α be a contracting ray in X. Step 1: Show f (α) stays bounded distance from some geodesic ray β in Y .

x

R quadratic in R !

y

Step 2: Show divβ(r) ≍ divα(r), hence α contracting ⇒ β contracting.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 15 / 18

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Contracting rays in CAT(0) cubical complexes

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 16 / 18

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Contracting rays in CAT(0) cubical complexes

Behrstock-C: Studied divergence for right-angled Artin group (RAAG). Introduced notion of “strongly separated walls” and showed: α has quadratic divergence ⇔ α crosses an infinite sequence of strongly separated walls.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 16 / 18

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Contracting rays in CAT(0) cubical complexes

Behrstock-C: Studied divergence for right-angled Artin group (RAAG). Introduced notion of “strongly separated walls” and showed: α has quadratic divergence ⇔ α crosses an infinite sequence of strongly separated walls.

Definition

Two walls H1, H2 in a CAT(0) cube complex X are strongly separated if H1 ∩ H2 = ∅ and no wall of X crosses both H1 and H2

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 16 / 18

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Contracting rays in CAT(0) cubical complexes

Behrstock-C: Studied divergence for right-angled Artin group (RAAG). Introduced notion of “strongly separated walls” and showed: α has quadratic divergence ⇔ α crosses an infinite sequence of strongly separated walls.

Definition

Two walls H1, H2 in a CAT(0) cube complex X are strongly separated if H1 ∩ H2 = ∅ and no wall of X crosses both H1 and H2 Caprace-Sageev: in very general CAT(0) cube complexes: X has a rank one isometry ⇔ X has a pair of strongly separated walls.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 16 / 18

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Want to characterize contracting rays in terms of strongly separated walls. First guess: A ray α is contracting⇔ it crosses an infinite sequence H1, H2, H3, . . . of strongly separated walls.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 17 / 18

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Want to characterize contracting rays in terms of strongly separated walls. First guess: A ray α is contracting⇔ it crosses an infinite sequence H1, H2, H3, . . . of strongly separated walls. Not quite!

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 17 / 18

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Want to characterize contracting rays in terms of strongly separated walls. First guess: A ray α is contracting⇔ it crosses an infinite sequence H1, H2, H3, . . . of strongly separated walls. Not quite! Too weak: if distance between Hi and Hi+1 is allowed to increase, α may stay longer and longer in a flat. So need d(hi, hi+1) < C where hi = α ∩ Hi.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 17 / 18

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Want to characterize contracting rays in terms of strongly separated walls. First guess: A ray α is contracting⇔ it crosses an infinite sequence H1, H2, H3, . . . of strongly separated walls. Not quite! Too weak: if distance between Hi and Hi+1 is allowed to increase, α may stay longer and longer in a flat. So need d(hi, hi+1) < C where hi = α ∩ Hi. Too strong: suppose α lies in a wall H. Then no two walls crossed by α are strongly separated.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 17 / 18

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Definition

Two walls H1, H2 in a CAT(0) cube complex X are k-separated if H1 ∩ H2 = ∅ and at most k walls of X cross both H1 and H2.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 18 / 18

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Definition

Two walls H1, H2 in a CAT(0) cube complex X are k-separated if H1 ∩ H2 = ∅ and at most k walls of X cross both H1 and H2. Assume ∃n such that at most n walls intersect the star of any vertex in X. (Probably stronger than necessary.)

Theorem

X as above. Then a geodesic ray α in X is contracting ⇔ ∃C > 0, k ∈ N, such that any segment of α of length C crosses a pair of k-separated walls.

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 18 / 18

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Definition

Two walls H1, H2 in a CAT(0) cube complex X are k-separated if H1 ∩ H2 = ∅ and at most k walls of X cross both H1 and H2. Assume ∃n such that at most n walls intersect the star of any vertex in X. (Probably stronger than necessary.)

Theorem

X as above. Then a geodesic ray α in X is contracting ⇔ ∃C > 0, k ∈ N, such that any segment of α of length C crosses a pair of k-separated walls.

Question

For CAT(0) cube complexes, what is the relation between ∂cX and the Poisson boundary described in Sageev’s talk?

Ruth Charney () Contracting Boundaries of CAT(0) Spaces Dubrovnik, July 2011 18 / 18