Contracting boundaries of CAT(0) spaces

Charney, Ruth; Sultan, Harold Mark

doi:10.1112/jtopol/jtu017

Cited by 105 publications

(205 citation statements)

References 30 publications

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“…The union of the other two edges is a

(1, L)

‐quasi‐geodesic and is hence contained in the

N^{'} false(1, L false)

‐neighborhood of the first edge, in fact, by [, Lemma 2.1], they have Hausdorff distance

2 N^{'} false(1, L false)

. By [, Lemma 2.5], this implies that the sides of

R

are

N^{''}

‐Morse where

N^{''}

depends only on

N^{'}, L

. Since

R

has vertices in

X

, it is

D_{N^{''}}

‐slim, and the sides of

P

that are contained in

T

, lie in the

2 N^{'} false(1, L false)

‐neighborhood of

R

.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section, we review some basic facts about Morse geodesics and define the Morse boundary. The reader is referred to for more details. Definition A geodesic

α

X

is Morse if there exists a function

N : R^{+} \times R^{+} \to R^{+}

such that any

(λ, ε)

‐quasi‐geodesic with endpoints on

α

, lies in the

N (λ, ε)

‐neighborhood of

α

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recall that a geodesic

α

D

‐contracting if the projection on

α

of any metric ball

B

, with

B \cap α = \emptyset

, has diameter at most

D

. It is shown in that in a CAT(0) space, given

D

, there exists

N

such that if

α

D

‐contracting, then it is

N

‐Morse, and conversely, given

N

there exists

D^{'}

such that if

α

N

‐Morse, then it is

D^{'}

contracting. It follows that for

X, Y

CAT(0),

f : \partial_{*} X \to \partial_{*} Y

is 2‐stable if and only if for each

D

there exists

D^{'}

such that

f

maps bi‐infinite

D

‐contracting geodesics to bi‐infinite

D^{'}

‐contracting geodesics.…”

Section: Homeomorphisms Induced By Quasi‐isometriesmentioning

confidence: 99%

“…Proof The fact that

\partial_{*} h

is a homeomorphism was proved by the first author and Sultan for CAT(0) spaces in and then generalized to arbitrary proper geodesic metric spaces by the second author in . The proof involves showing that for each Morse guage

N

, there exists a Morse guage

N^{'}

such that the image of an

N

‐Morse ray under the quasi‐isometry

h

can be ‘straightened’ to an

N^{'}

‐Morse ray in

Y

.…”

Section: Homeomorphisms Induced By Quasi‐isometriesmentioning

confidence: 99%

“…We denote the Morse boundary of

X

\partial_{*} X

. The key property of this boundary is quasi‐isometry invariance; a quasi‐isometry between two proper geodesic metric spaces induces a homeomorphism on their Morse boundaries . Thus, the Morse boundary is well defined for any finitely generated group (though it may be empty if the group has no Morse geodesics).…”

Section: Introductionmentioning

confidence: 99%

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Quasi‐Mobius Homeomorphisms of Morse boundaries

Charney

Cordes

Murray

2019

Bull. London Math. Soc.

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The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic‐like behavior in the space. A key property of this boundary is that a quasi‐isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper, we investigate when the converse holds. We prove that for X,Y proper, cocompact spaces, a homeomorphism between their Morse boundaries is induced by a quasi‐isometry if and only if the homeomorphism is quasi‐mobius and 2‐stable.

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“…The union of the other two edges is a

(1, L)

‐quasi‐geodesic and is hence contained in the

N^{'} false(1, L false)

‐neighborhood of the first edge, in fact, by [, Lemma 2.1], they have Hausdorff distance

2 N^{'} false(1, L false)

. By [, Lemma 2.5], this implies that the sides of

R

are

N^{''}

‐Morse where

N^{''}

depends only on

N^{'}, L

. Since

R

has vertices in

X

, it is

D_{N^{''}}

‐slim, and the sides of

P

that are contained in

T

, lie in the

2 N^{'} false(1, L false)

‐neighborhood of

R

.…”

Section: Preliminariesmentioning

confidence: 99%

“…In this section, we review some basic facts about Morse geodesics and define the Morse boundary. The reader is referred to for more details. Definition A geodesic

α

X

is Morse if there exists a function

N : R^{+} \times R^{+} \to R^{+}

such that any

(λ, ε)

‐quasi‐geodesic with endpoints on

α

, lies in the

N (λ, ε)

‐neighborhood of

α

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Recall that a geodesic

α

D

‐contracting if the projection on

α

of any metric ball

B

, with

B \cap α = \emptyset

, has diameter at most

D

. It is shown in that in a CAT(0) space, given

D

, there exists

N

such that if

α

D

‐contracting, then it is

N

‐Morse, and conversely, given

N

there exists

D^{'}

such that if

α

N

‐Morse, then it is

D^{'}

contracting. It follows that for

X, Y

CAT(0),

f : \partial_{*} X \to \partial_{*} Y

is 2‐stable if and only if for each

D

there exists

D^{'}

such that

f

maps bi‐infinite

D

‐contracting geodesics to bi‐infinite

D^{'}

‐contracting geodesics.…”

Section: Homeomorphisms Induced By Quasi‐isometriesmentioning

confidence: 99%

“…Proof The fact that

\partial_{*} h

N

, there exists a Morse guage

N^{'}

such that the image of an

N

‐Morse ray under the quasi‐isometry

h

can be ‘straightened’ to an

N^{'}

‐Morse ray in

Y

.…”

Section: Homeomorphisms Induced By Quasi‐isometriesmentioning

confidence: 99%

“…We denote the Morse boundary of

X

\partial_{*} X

Section: Introductionmentioning

confidence: 99%

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Quasi‐Mobius Homeomorphisms of Morse boundaries

Charney

Cordes

Murray

2019

Bull. London Math. Soc.

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Square percolation and the threshold for quadratic divergence in random right‐angled Coxeter groups

Behrstock

Falgas–Ravry

Susse

2021

Random Struct Algorithms

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Given a graph Γ, its auxiliary square‐graph □(Γ) is the graph whose vertices are the non‐edges of Γ and whose edges are the pairs of non‐edges which induce a square (i.e., a 4‐cycle) in Γ. We determine the threshold edge‐probability p=pc(n) at which the Erdős–Rényi random graph Γ=Γn,p begins to asymptotically almost surely (a.a.s.) have a square‐graph with a connected component whose squares together cover all the vertices of Γn,p. We show pc(n)=6−2/n, a polylogarithmic improvement on earlier bounds on pc(n) due to Hagen and the authors. As a corollary, we determine the threshold p=pc(n) at which the random right‐angled Coxeter group WΓn,p a.a.s. becomes strongly algebraically thick of order 1 and has quadratic divergence.

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