# Groups, Geometry, and Dynamics

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**Volume 8, Issue 4, 2014, pp. 1047–1099**

**DOI: 10.4171/GGD/256**

Published online: 2014-12-31

The limit set of subgroups of arithmetic groups in $\mathrm{PSL} (2, \mathbb C)^q \times \mathrm{PSL} (2, \mathbb R)^r$

Slavyana Geninska^{[1]}(1) Université Paul Sabatier, Toulouse, France

We consider subgroups $\Gamma$ of arithmetic groups in the product $\mathrm{PSL} (2,\mathbb C)^q\times \mathrm {PSL}(2,\mathbb R)^r$ with $q+r\geq 2$ and their limit set. We prove that the projective limit set of a nonelementary finitely generated $\Gamma$ consists of exactly one point if and only if one and hence all projections of $\Gamma$ to the simple factors of $\mathrm {PSL}(2,\mathbb C)^q\times \mathrm {PSL}(2,\mathbb R)^r$ are subgroups of arithmetic Fuchsian or Kleinian groups. Furthermore, we study the topology of the whole limit set of $\Gamma$. 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 $\Gamma$ with its arithmetic ones.

*Keywords: *Fuchsian groups, arithmetic lattices, limit sets

Geninska Slavyana: The limit set of subgroups of arithmetic groups in $\mathrm{PSL} (2, \mathbb C)^q \times \mathrm{PSL} (2, \mathbb R)^r$. *Groups Geom. Dyn.* 8 (2014), 1047-1099. doi: 10.4171/GGD/256