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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Discrete subgroups of the special linear group with thin limit sets
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by Aaram Yun PDF
Trans. Amer. Math. Soc. 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