Discrete subgroups of the special linear group with thin limit sets (Q2826763)

For a discrete subgroup \(\Gamma\subset \mathrm{SL}(n+1,{\mathbb R})\) one can consider its limit set in the projective space \({\mathbb P}^n\) defined as follows. A sequence \(g_k \in \mathrm{SL}(n+ 1,{\mathbb R})\) is said to be contracting if \(\frac{g_k}{\| g_k\|}\) converges to a linear map \(\delta \) of rank one. In this case the image of \(\delta\) is a point in \({\mathbb P}^n\) and is called the limit point of the sequence. The limit set \(\Lambda(\Gamma)\subset{\mathbb P}^n\) is the set of all limit points.NEWLINENEWLINEThe author shows on the one hand that the limit set of a Zariski-dense, discrete subgroup \(\Gamma\subset \mathrm{SL}(n+1,{\mathbb R})\) can not be contained in a \(C^\infty\)-smooth curve. Moreover, for a Zariski-dense, discrete subgroup \(\Gamma\subset \mathrm{SL}(3,{\mathbb R})\) the limit set can not even be contained in a \(C^2\)-smooth curve. These results can be compared with similar results for Kleinian groups, where a non-elementary, discrete group with limit set on a \(C^1\)-smooth curve must actually have its limit set on a circle, which yields a Zariski-closed condition.NEWLINENEWLINEThe main result of the paper is however, for any \(n\geq 3\) and \(N\in{\mathbb N}\), the construction of Zariski-dense discrete subgroups \(\Gamma\subset\mathrm{SL}(n+1,{\mathbb R})\) whose limit set is contained in a \(C^N\)-smooth curve. This is achieved by considering a free group over a specially chosen generating set \(S\), and applying the ping-pong lemma to the action of this free group on the \(N\)-th order jet bundle over \({\mathbb P}^n\).NEWLINENEWLINEThe specially chosen generating sets \(S\) have to satisfy certain technical conditions (``thinness'') which are motivated by the classical case of Schottky groups in \(\mathrm{SL}(2,{\mathbb R})\), but which are too involved to be described here in any detail. (And so are the proofs.) In the words of the author, the main point is to control various \(C^N\)-smooth curve segments and their images by elements \(g\in\Gamma\). These curve segments are then used for constructing the curve containing \(\Lambda(\Gamma)\).