Hyperbolic quasi-geodesics in CAT(0) spaces

Sultan, Harold Mark

doi:10.1007/s10711-013-9851-4

Cited by 31 publications

(37 citation statements)

References 15 publications

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“…We are now ready to prove the equivalence of the contracting and Morse conditions. We remark that, in [30], the first author proved that these two notions are equivalent, but without the explicit control on the constants. This control on constants will be essential to our understanding of contracting boundaries.…”

Section: Contracting and Morse Geodesicsmentioning

confidence: 98%

“…The fact that (1) =⇒ (2) with explicit constants, is the well-known 'Morse stability lemma'. For a proof see, for instance, [1] or [30].…”

Section: Moreover the Morse Function M Is Determined By The Constantmentioning

confidence: 99%

“…The fact that (1) implies (2) is a well-known result, an explicit proof of which is given by Algom-Kfir [1]. In [30], the second author shows that (2) implies (1), but without explicit control on the constants. In [5], Bestvina and Fujiwara develop many properties of contracting geodesics and, in particular, prove that (1) implies (3).…”

Section: Introductionmentioning

confidence: 97%

“…There are various well-studied properties that can be used to define precise notions of 'hyperbolic type' geodesics including the Morse property, the contracting property, superlinear divergence, and slimness (see Section 2 for definitions). These notions have proved fruitful in analyzing right-angled Artin groups (RAAGs) [3], Teichmüller space [2,[7][8][9], the mapping class group [2], CAT(0) spaces [4,5,30], and Out(F n ) [1] among others (see also [13,14,21,22]).…”

Section: Introductionmentioning

confidence: 99%

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Contracting boundaries of CAT(0) spaces

Charney

Sultan

2014

Journal of Topology

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105

201

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As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well‐defined since quasi‐isometric CAT(0) spaces can have non‐homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up of rays satisfying one of five hyperbolic‐like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi‐isometry invariant. We use this invariant to distinguish the quasi‐isometry classes of certain right‐angled Coxeter groups.

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Section: Contracting and Morse Geodesicsmentioning

confidence: 98%

“…The fact that (1) =⇒ (2) with explicit constants, is the well-known 'Morse stability lemma'. For a proof see, for instance, [1] or [30].…”

Section: Moreover the Morse Function M Is Determined By The Constantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Contracting boundaries of CAT(0) spaces

Charney

Sultan

2014

Journal of Topology

Self Cite

105

201

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“…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%

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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A combination theorem for Anosov subgroups

2018

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We prove an analogue of Klein combination theorem for Anosov subgroups by using a localto-global principle for Morse quasigeodesics. arXiv:1805.07374v2 [math.GR] 4 Jul 2018 free products and HNN extensions. These so called "Klein-Maskit combination theorems" have been generalized to the geometrically finite subgroups of the isometry groups of higher dimensional hyperbolic spaces by several mathematicians. For instance, in [BC08], Baker and Cooper proved the following theorem.Theorem 1.1 (Virtual amalgam theorem, [BC08]). If Γ 1 and Γ 2 are two geometrically finite subgroups of Isom (H n ) which have compatible parabolic subgroups, and if H = Γ 1 ∩Γ 2 is separable in Γ 1 and Γ 2 , then there exists finite index subgroups Γ 1 and Γ 2 of Γ 1 and Γ 2 , respectively, containing H such that the group Γ 1 , Γ 2 generated by Γ 1 and Γ 2 is geometrically finite, and is naturally isomorphic to the amalgam Γ 1 * H Γ 2 .When Γ 1 and Γ 2 intersect trivially, the "compatibility condition" in the above theorem simply means that the limit sets of Γ 1 and Γ 2 in ∂ ∞ H n are disjoint. Since this case would be most relevant to our work, we state it separately.Corollary 1.2. If Γ 1 and Γ 2 are two geometrically finite subgroups of Isom (H n ) with disjoint limit sets in ∂ ∞ H n , then there exists finite index subgroups Γ 1 and Γ 2 of Γ 1 and Γ 2 , respectively, such that the group Γ 1 , Γ 2 generated by Γ 1 and Γ 2 is geometrically finite and is naturally isomorphicThere are also certain generalizations of these combination theorems in the realm of subgroups of hyperbolic groups and, more generally, isometry groups of Gromov-hyperbolic spaces. In [Git99], Gitik proved that under certain conditions two quasiconvex subgroups of a δ-hyperbolic group "can be virtually amalgamated." In this regard, our main theorem is an analogue of [Git99, Corollary 3]. See also the papers by Martínez-Pedroza [MP09] and Martínez-Pedroza-Sisto [MPS12] for closely related results.In the present work, we prove a combination theorem for Anosov subgroups of semisimple Lie groups. Anosov representations of surface groups (and, more generally, fundamental groups of compact negatively curved manifolds) were introduced by Labourie [Lab06] to study the "Hitchin component" of the space of reducible representations in P SL(n, R). Guichard and Weinhard [GW12] generalized this notion in the setting of representations of hyperbolic groups in real semisimple Lie groups. Anosov subgroups can be regarded as higher rank generalizations of convex-cocompact subgroups of isometry groups of negatively curved symmetric spaces.Our main result presents an analogue of Corollary 1.2 for Anosov subgroups. Before stating our theorem, we briefly discuss our framework. Let G be a semisimple Lie group, let P be a maximal parabolic subgroup conjugate to its opposite subgroups.Our main result is the following.

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Hyperbolic quasi-geodesics in CAT(0) spaces

Cited by 31 publications

References 15 publications

Contracting boundaries of CAT(0) spaces

Contracting boundaries of CAT(0) spaces

Quasi‐Mobius Homeomorphisms of Morse boundaries

A combination theorem for Anosov subgroups

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