Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II

1997 ◽  
Vol 7 (3) ◽  
pp. 561-593 ◽  
Author(s):  
Z. Sela
Keyword(s):  
Lie Groups ◽  
Hyperbolic Groups ◽  
Discrete Groups ◽  
Gromov Hyperbolic

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 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.

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 Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space

2020 ◽  
Vol 14 (2) ◽  
pp. 369-411
Author(s):  
Katsuhiko Matsuzaki ◽  
Yasuhiro Yabuki ◽  
Johannes Jaerisch
Keyword(s):  
Hyperbolic Space ◽  
Discrete Groups ◽  
Gromov Hyperbolic ◽  
Divergence Type

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 Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

2019 ◽  
Vol 2019 (757) ◽  
pp. 197-246 ◽  
Author(s):  
Daniel Drimbe ◽  
Daniel Hoff ◽  
Adrian Ioana
Keyword(s):  
Lie Groups ◽  
Hyperbolic Groups ◽  
Arithmetic Groups ◽  
Semisimple Lie Groups ◽  
Infinite Groups ◽  
Prime Factorization ◽  
Finite Center ◽  
Simple Lie Groups ◽  
Rank One ◽  
Higher Rank

AbstractWe prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the {\mathrm{II}_{1}} factor {L(\Gamma)} is prime. In particular, we deduce that the {\mathrm{II}_{1}} factors associated to the arithmetic groups {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer {d\geq 2} with {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for {L(\Gamma)}.

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