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Strongly dense representations of surface groups

Part of: Other groups of matrices

Published online by Cambridge University Press:  27 January 2025

D. D. Long  and
A. W. Reid Open the ORCID record for A. W. Reid [Opens in a new window]
D. D. Long
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA, USA
A. W. Reid*
Affiliation:
Department of Mathematics, Rice University, Houston, TX, USA
*
Corresponding author: A. W. Reid, email: alan.reid@rice.edu
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Abstract

The notion of a strongly dense subgroup was introduced by Breuillard, Green, Guralnick and Tao: a subgroup Γ of a semi-simple $\mathbb{Q}$ algebraic group $\mathcal{G}$ is called strongly dense if every pair of non-commuting elements generate a Zariski dense subgroup. Amongst other things, Breuillard et al. prove that there exist strongly dense free subgroups in $\mathcal{G}({\mathbb{R}})$ and ask whether or not a Zariski dense subgroup of $\mathcal{G}(\mathbb{R})$ always contains a strongly dense free subgroup. In this paper, we answer this for many surface subgroups of $\textrm{SL}(3,\mathbb{R})$.

Keywords

triangle groupHitchin representationstrongly densesubgroup

MSC classification

Secondary: 20H10: Fuchsian groups and their generalizations

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 68 , Issue 2 , May 2025 , pp. 583 - 593
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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Footnotes

†

Both authors partially supported by the NSF

References

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