The limit set of subgroups of arithmetic groups in PSL(2, C)q x PSL(2, R)r

We consider subgroups Gamma of arithmetic groups in the product PSL(2, C)(q) x PSL(2, R)(r) with q + r >= 2 and their limit set. We prove that the projective limit set of a nonelementary finitely generated. consists of exactly one point if and only if one and hence all projections of. to the simple factors of PSL(2, C)(q) x PSL(2, 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. with its arithmetic ones.