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Discrete subgroups of the special linear group with thin limit sets


Author: Aaram Yun
Journal: Trans. Amer. Math. Soc. 369 (2017), 365-407
MSC (2010): Primary 22E40
DOI: https://doi.org/10.1090/tran/6753
Published electronically: May 2, 2016
MathSciNet review: 3557777
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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

DOI: https://doi.org/10.1090/tran/6753
Received by editor(s): November 26, 2012
Received by editor(s) in revised form: December 27, 2014
Published electronically: May 2, 2016
Article copyright: © Copyright 2016 American Mathematical Society