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

Abstract

We consider subgroups $\Gamma$ of arithmetic groups in the product $\mathrm{PSL}(2,\mathbb{C}{)}^q\times \mathrm{PSL}(2,\mathbb{R}{)}^r$ with $q+r\ge 2$ 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.