scholarly journals Conformal dimension and Gromov hyperbolic groups with 2–sphere boundary

2005 ◽  
Vol 9 (1) ◽  
pp. 219-246 ◽  
Author(s):  
Mario Bonk ◽  
Bruce Kleiner
Keyword(s):  
Hyperbolic Groups ◽  
Conformal Dimension ◽  
Gromov Hyperbolic

scholarly journals Conformal dimension of hyperbolic groups that split over elementary subgroups

2021 ◽  
Author(s):  
Matias Carrasco ◽  
John M. Mackay
Keyword(s):  
Hyperbolic Groups ◽  
Conformal Dimension ◽  
Gromov Hyperbolic

AbstractWe study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without 2-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.

Growth of conjugacy classes in Gromov hyperbolic groups

2002 ◽  
Vol 12 (3) ◽  
pp. 464-478 ◽  
Author(s):  
M. Coornaert ◽  
G. Knieper
Keyword(s):  
Conjugacy Classes ◽  
Hyperbolic Groups ◽  
Gromov Hyperbolic

scholarly journals Horofunctions and symbolic dynamics on Gromov hyperbolic groups

2001 ◽  
Vol 43 (03) ◽  
Author(s):  
Michel Coornaert ◽  
Athanase Papadopoulos
Keyword(s):  
Symbolic Dynamics ◽  
Hyperbolic Groups ◽  
Gromov Hyperbolic

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 Gromov hyperbolic groups and the Macaev norm

2006 ◽  
Vol 223 (1) ◽  
pp. 141-157 ◽  
Author(s):  
Rui Okayasu
Keyword(s):  
Hyperbolic Groups ◽  
Gromov Hyperbolic

scholarly journals Cross ratios and cubulations of hyperbolic groups

2021 ◽  
Author(s):  
Jonas Beyrer ◽  
Elia Fioravanti
Keyword(s):  
Length Spectrum ◽  
Hyperbolic Groups ◽  
Optimal Length ◽  
Rigidity Result ◽  
Cube Complex ◽  
Geodesic Currents ◽  
Gromov Hyperbolic ◽  
Cube Complexes ◽  
The Relationship ◽  
Gromov Boundary

AbstractMany geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Sign in / Sign up

Export Citation Format

Share Document