## Discrete subgroups of the special linear group with thin limit sets

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**369**(2017), 365-407 Request permission

## 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.

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## Additional Information

**Aaram Yun**- Affiliation: School of Electrical & Computer Engineering, Ulsan National Institute of Science and Technology (UNIST), Ulsan, Korea
- Email: aaramyun@unist.ac.kr
- Received by editor(s): November 26, 2012
- Received by editor(s) in revised form: December 27, 2014
- Published electronically: May 2, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 365-407 - MSC (2010): Primary 22E40
- DOI: https://doi.org/10.1090/tran/6753
- MathSciNet review: 3557777