Discrete subgroups of the special linear group with thin limit sets

Abstract:

In this paper, we construct a discrete Zariski-dense subgroup $\Gamma$ of $\mathrm {SL}(n+1,\mathbb {R})$ whose limit set on $\mathbb {P}^{n}$ is ‘thin’, that is, contained in a $C^N$-smooth curve, for any $n\geq 3$ 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 $\mathbb {P}^{n}$.

We also show that in a sense this is the best possible result: we show that there does not exist any Zariski-dense subgroup $\Gamma \subseteq \mathrm {SL}(3,\mathbb {R})$ whose limit set is contained in a $C^{2}$-smooth curve, and there does not exist any Zariski-dense subgroup $\Gamma \subseteq \mathrm {SL}(n+1,\mathbb {R})$ whose limit set is contained in a $C^\infty$-smooth curve.