Geodesic
On geometric convergence of discrete groups
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 305-310 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\{G_{j}\}$ of ${\rm SO}(n,1)$ or ${\rm PU}(n,1)$, if $\{G_{j}\}$ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.
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 $\{G_{j}\}$ of ${\rm SO}(n,1)$ or ${\rm PU}(n,1)$, if $\{G_{j}\}$ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.
DOI : 10.1007/s10587-014-0102-0
Classification : 20H10, 30C62, 30F40
Keywords: discrete group; geometric convergence; uniformly bounded torsion 
@article{10_1007_s10587_014_0102_0,
     author = {Yang, Shihai},
     title = {On geometric convergence of discrete groups},
     journal = {Czechoslovak Mathematical Journal},
     pages = {305--310},
     year = {2014},
     volume = {64},
     number = {2},
     doi = {10.1007/s10587-014-0102-0},
     mrnumber = {3277737},
     zbl = {06391495},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0102-0/}
}
TY  - JOUR
AU  - Yang, Shihai
TI  - On geometric convergence of discrete groups
JO  - Czechoslovak Mathematical Journal
PY  - 2014
SP  - 305
EP  - 310
VL  - 64
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0102-0/
DO  - 10.1007/s10587-014-0102-0
LA  - en
ID  - 10_1007_s10587_014_0102_0
ER  - 
%0 Journal Article
%A Yang, Shihai
%T On geometric convergence of discrete groups
%J Czechoslovak Mathematical Journal
%D 2014
%P 305-310
%V 64
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0102-0/
%R 10.1007/s10587-014-0102-0
%G en
%F 10_1007_s10587_014_0102_0
Yang, Shihai. On geometric convergence of discrete groups. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 305-310. doi: 10.1007/s10587-014-0102-0

Cité par Sources :