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

TL;DR

This paper characterizes the limit sets of certain subgroups of arithmetic groups in products of PSL(2,C) and PSL(2,R), showing conditions under which the limit set is a single point or homeomorphic to a circle.

Contribution

It provides a new classification of limit sets for subgroups of arithmetic groups in product Lie groups, linking their structure to projections onto simple factors.

Findings

The projective limit set is a single point iff all projections are arithmetic Fuchsian or Kleinian groups.

Conditions are established for the limit set to be homeomorphic to a circle.

The topology of the limit set depends on the projections of the subgroup.

Abstract

While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider subgroups $\Gamma$ of arithmetic groups in $PSL(2,C)^q \times PSL(2,R)^r$ with $q+r>1$ 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 $PSL(2,C)^q \times PSL(2,R)^r$ 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.