Discrete subgroups of the special linear group with thin limit sets

In this paper, we construct a discrete Zariski-dense subgroup Γ of SL(n+1,R) whose limit set on Pⁿ 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 Pⁿ. We also show that in a sense this is the best possible result: we show that there does not exist any Zariski-dense subgroup Γ ⊆ SL(3,R) whose limit set is contained in a C²-smooth curve, and there does not exist any Zariski-dense subgroup Γ ⊆ SL(n+1,R) whose limit set is contained in a C^∞-smooth curve.