scholarly journals Random walks and quasi‐convexity in acylindrically hyperbolic groups

2021 ◽  
Vol 14 (3) ◽  
pp. 992-1026
Author(s):  
Carolyn Abbott ◽  
Michael Hull
Keyword(s):  
Random Walks ◽  
Hyperbolic Groups ◽  
Quasi Convexity

Random walks on three-strand braids and on related hyperbolic groups

2002 ◽  
Vol 36 (1) ◽  
pp. 43-66 ◽  
Author(s):  
Sergei Nechaev ◽  
Rapha l Voituriez
Keyword(s):  
Random Walks ◽  
Hyperbolic Groups

scholarly journals Random walks on weakly hyperbolic groups

2018 ◽  
Vol 2018 (742) ◽  
pp. 187-239 ◽  
Author(s):  
Joseph Maher ◽  
Giulio Tiozzo
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Linear Growth ◽  
Countable Group ◽  
Convergence Result ◽  
Hyperbolic Groups ◽  
Poisson Boundary ◽  
Gromov Hyperbolic ◽  
Gromov Boundary ◽  
Translation Length

Abstract Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

scholarly journals Random Subgroups of Acylindrically Hyperbolic Groups and Hyperbolic Embeddings

2017 ◽  
Vol 2019 (13) ◽  
pp. 3941-3980 ◽  
Author(s):  
Joseph Maher ◽  
Alessandro Sisto
Keyword(s):  
Normal Subgroup ◽  
Random Walks ◽  
Free Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Finite Collection ◽  
Small Cancellation ◽  
Cancellation Condition ◽  
Asymptotic Probability ◽  
Small Cancellation Condition

Abstract Let $G$ be an acylindrically hyperbolic group. We consider a random subgroup $H$ in $G$, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup $H$ of $G$ is a free group, and the semidirect product of $H$ acting on $E(G)$ is hyperbolically embedded in $G$, where $E(G)$ is the unique maximal finite normal subgroup of $G$. Furthermore, with control on the lengths of the generators, we show that $H$ satisfies a small cancellation condition with asymptotic probability one.

scholarly journals Local limit theorems in relatively hyperbolic groups I: rough estimates

2021 ◽  
pp. 1-41
Author(s):  
MATTHIEU DUSSAULE
Keyword(s):  
Green Function ◽  
Random Walks ◽  
Limit Theorems ◽  
Local Limit ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
The Green Function ◽  
Positive Recurrent ◽  
Relatively Hyperbolic ◽  
Local Limit Theorems

Abstract This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.

scholarly journals Regularity of the entropy for random walks on hyperbolic groups

2013 ◽  
Vol 41 (5) ◽  
pp. 3582-3605 ◽  
Author(s):  
François Ledrappier
Keyword(s):  
Random Walks ◽  
Hyperbolic Groups

scholarly journals A Numerical Lower Bound for the Spectral Radius of Random Walks on Surface Groups

2015 ◽  
Vol 24 (6) ◽  
pp. 838-856 ◽  
Author(s):  
S. GOUEZEL
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Spectral Radius ◽  
Cayley Graphs ◽  
Surface Group ◽  
Hyperbolic Groups ◽  
Surface Groups ◽  
Genus 2 ◽  
Order Of Magnitude

Estimating numerically the spectral radius of a random walk on a non-amenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.

Sign in / Sign up

Export Citation Format

Share Document