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While lattices in semi-simple Lie groups are studied very well, only littleis known about discrete subgroups of infinite covolume. The main class ofexamples are Schottky groups. Here we investigate some new examples.  We consider subgroups $\Gamma$ of arithmetic groups in $PSL(2,C)^q \timesPSL(2,R)^r$ with $q+r>1$ and their limit set. We prove that the projectivelimit set of a nonelementary finitely generated $\Gamma$ consists of exactlyone point if and only if one and hence all projections of $\Gamma$ to thesimple factors of $PSL(2,C)^q \times PSL(2,R)^r$ are subgroups of arithmeticFuchsian or Kleinian groups. Furthermore, we study the topology of the wholelimit set of $\Gamma$. In particular, we give a necessary and sufficientcondition for the limit set to be homeomorphic to a circle.

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