On geometric convergence of discrete groups

Yang, Shihai. "On geometric convergence of discrete groups." Czechoslovak Mathematical Journal 64.2 (2014): 305-310. <http://eudml.org/doc/262055>.

@article{Yang2014,
abstract = {One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if $\Gamma $ is a non-elementary finitely generated group and $\rho _\{i\}\colon \Gamma \rightarrow \{\rm SO\}(n,1)$ a sequence of discrete and faithful representations, then the geometric limit of $\rho _\{i\}(\Gamma )$ is a discrete subgroup of $\{\rm SO\}(n,1)$. We generalize this result by showing that for a sequence of discrete and non-elementary subgroups $\lbrace G_\{j\}\rbrace $ of $\{\rm SO\}(n,1)$ or $\{\rm PU\}(n,1)$, if $\lbrace G_\{j\}\rbrace $ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.},
author = {Yang, Shihai},
journal = {Czechoslovak Mathematical Journal},
keywords = {discrete group; geometric convergence; uniformly bounded torsion; discrete group; hyperbolic space; geometric convergence; uniformly bounded torsion},
language = {eng},
number = {2},
pages = {305-310},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On geometric convergence of discrete groups},
url = {http://eudml.org/doc/262055},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Yang, Shihai
TI - On geometric convergence of discrete groups
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 305
EP - 310
AB - One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if $\Gamma $ is a non-elementary finitely generated group and $\rho _{i}\colon \Gamma \rightarrow {\rm SO}(n,1)$ a sequence of discrete and faithful representations, then the geometric limit of $\rho _{i}(\Gamma )$ is a discrete subgroup of ${\rm SO}(n,1)$. We generalize this result by showing that for a sequence of discrete and non-elementary subgroups $\lbrace G_{j}\rbrace $ of ${\rm SO}(n,1)$ or ${\rm PU}(n,1)$, if $\lbrace G_{j}\rbrace $ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.
LA - eng
KW - discrete group; geometric convergence; uniformly bounded torsion; discrete group; hyperbolic space; geometric convergence; uniformly bounded torsion
UR - http://eudml.org/doc/262055
ER -