Discrete subgroups of the special linear group with thin limit sets

Abstract

In this paper, we construct a discrete Zariski-dense subgroup Gamma of SL(n+1, R) whose limit set on P-n is 'thin', that is, contained in a C-N-smooth curve, for any n >= 3 and N > 0. We achieve this by applying the ping-pong lemma to the action of a specially chosen generating set S on the N-th order jet bundle over P-n.

We also show that in a sense this is the best possible result: we show that there does not exist any Zariski-dense subgroup Gamma subset of SL(3, R) whose limit set is contained in a C-2-smooth curve, and there does not exist any Zariski-dense subgroup Gamma subset of SL(n+1, R) whose limit set is contained in a C-infinity-smooth curve.e.