# The limit set of subgroups of arithmetic groups in $PSL(2,C)_{q}×PSL(2,R)_{r}$

### Slavyana Geninska

Université Paul Sabatier, Toulouse, France

## Abstract

We consider subgroups $Γ$ of arithmetic groups in the product $PSL(2,C)_{q}×PSL(2,R)_{r}$ with $q+r≥2$ and their limit set. We prove that the projective limit set of a nonelementary finitely generated $Γ$ consists of exactly one point if and only if one and hence all projections of $Γ$ to the simple factors of $PSL(2,C)_{q}×PSL(2,R)_{r}$ are subgroups of arithmetic Fuchsian or Kleinian groups. Furthermore, we study the topology of the whole limit set of $Γ$. In particular, we give a necessary and sufficient condition for the limit set to be homeomorphic to a circle. This result connects the geometric properties of $Γ$ with its arithmetic ones.

## Cite this article

Slavyana Geninska, The limit set of subgroups of arithmetic groups in $PSL(2,C)_{q}×PSL(2,R)_{r}$. Groups Geom. Dyn. 8 (2014), no. 4, pp. 1047–1099

DOI 10.4171/GGD/256