Contracting boundaries of CAT(0) spaces

By Ruth Charney, Harold Sultan. 2013

133 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c889da3bf66f4cebc4681dee390bbec2b25ee963

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In Applied Mathematics

1. Christopher H. Cashen
2. Goulnara Arzhantseva
3. Matthew Cordes
4. David Hume
5. Dominik Gruber
6. Hung Cong Tran
7. Wen-yuan Yang
8. Anthony Genevois

In Geometry and Topology

1. Hung Cong Tran
2. Wen-yuan Yang
3. Dominik Gruber
4. Christopher H. Cashen
5. Goulnara Arzhantseva
6. David Hume
7. Anthony Genevois
8. Matthew Cordes

Citations in the corpus

• On strongly quasiconvex subgroups (2017, background): “In [CS15], Charney-Sultan introduce the concept of lower divergence of quasi-geodesic to study Morse quasi-geodesics in CAT(0) spaces.”
• Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction (2016, methodology): “If X is CAT(0) and γ is a geodesic then this is a recent theorem of Charney and Sultan [13].”
• Morse Boundaries of Proper Geodesic Metric Spaces (2015, background): “s1,A] is bounded by 2N(1,4N(3,0)) and thus the Hausdorff distance between γ[0,x] and α1[s1,A] is bounded by 2N(1,4N(3,0))+ 4N(3,0). We get identical bounds for γ[x,C] and α2[s2,B]. By Lemma 2.5 (1) of [3] they are N′′-Morse where N′′ depends on N. Let σ: [a,b] →Xbe a (λ,ǫ) quasi-geodesic with endpoin”
• Hyperbolicities in CAT(0) cube complexes (2017, unknown): “In order to find a geometric characterization, we need Charney and Sultan’s contracting boundary [CS15].”
• Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups (2016, methodology): “In [CS15], Charney and Sultan used contracting isometries to distinguish quasi-isometrically some cubulable groups, and they introduced k-separated hyperplanes in order to characterize contracting isometries in uniformly locally finite CAT(0) cube complexes [CS15, Theorem 4.”
• Statistically convex-cocompact actions of groups with contracting elements (2016, background): “er" statement follows by [28, Lemma 4.3]. Remark. The above proof of Lemma 4.2 still works for the set V;P(n; )replaced by A(o;n; ): in fact, the proof gets greatly simplied. So the inequality (21) holds also for a!(n; ) ∶= exp(−!n)⋅♯A(o;n; ). See the proof of Theorem 5.3 below. 4.2. The main c”