Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 22E40

Retrieve articles in all journals with MSC (2010): 22E40


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
Article copyright: © Copyright 2016 American Mathematical Society