Statistically Convex-Cocompact Actions of Groups with Contracting Elements

Yang, Wenyuan

doi:10.1093/imrn/rny001

Cited by 20 publications

(68 citation statements)

References 78 publications

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“…Note. While putting the finishing touches on this paper, we learned about the existence of the preprint [Yang, 2016], which prove a more general result regarding the genericity of contracting elements of groups. The proofs and techniques explained in this paper, which mainly use Garside theory, have been done independently and simultaneously with Yang's article.…”

Section: Proof Of Theoremmentioning

confidence: 92%

On loxodromic actions of Artin–Tits groups

Cumplido

2019

Journal of Pure and Applied Algebra

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Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that "most" elements of Artin-Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin-Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin-Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.

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Section: Proof Of Theoremmentioning

confidence: 92%

On loxodromic actions of Artin–Tits groups

Cumplido

2019

Journal of Pure and Applied Algebra

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“…We need some preliminary results. We refer to [40] for definitions and background for statistically convex co-compact group actions and contracting elements. For our purposes, we will only use that such actions Γ (X, d) include the actions of groups with infinite Floyd boundary on their Cayley graphs [39,Lemma 7.2].…”

Section: Deviation Inequalitiesmentioning

confidence: 99%

Stability phenomena for Martin boundaries of relatively hyperbolic groups

Dussaule,

Gekhtman

2019

Preprint

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Let Γ be a relatively hyperbolic group and let µ be an admissible symmetric finitely supported probability measure on Γ. We extend Floyd-Ancona type inequalities from [11] up to the spectral radius of µ. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on Γ is stable in the sense of Picardello and Woess [28]. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfied in small rank. so that in particular, the Martin boundary of an admissible symmetric finitely supported probability measure on a geometrically finite Kleinian group of dimension at most 5 is always strongly stable.This can be written as P f = rf , where P is the Markov operator associated with µ. Every r-harmonic positive function can be represented as an integral over the Martin boundary. Precisely, for every such function f , there exists a probability measure ν f on ∂ rµ Γ such that for all x ∈ Γ,In general, the measure ν f is not unique. To obtain uniqueness, we restrict our attention to the minimal boundary that we now define.

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“…We say that G X has complementary growth gap if the complementary growth is strictly less than δ G . Yang [25] proved that if G acts properly with a strongly contracting element and 0 < δ G < ∞ then complementary growth gap implies purely exponential growth.…”

Section: Preliminariesmentioning

confidence: 99%

Cogrowth for group actions with strongly contracting elements

Arzhantseva¹,

Cashen²

2018

Ergod. Th. Dynam. Sys.

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Let G be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let N be an infinite normal subgroup of G, and let δ N and δ G be the growth rates of N and G with respect to the pseudo-metric induced by the action. We prove that if G has purely exponential growth with respect to the pseudo-metric then δ N /δ G > 1/2. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk's original result on free groups with respect to a word metrics and a recent result of Jaerisch, Matsuzaki, and Yabuki on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

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Statistically Convex-Cocompact Actions of Groups with Contracting Elements

Cited by 20 publications

References 78 publications

On loxodromic actions of Artin–Tits groups

On loxodromic actions of Artin–Tits groups

Stability phenomena for Martin boundaries of relatively hyperbolic groups

Cogrowth for group actions with strongly contracting elements

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