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Strongly dense representations of surface groups
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 27 January 2025, pp. 1-11
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- Article
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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 
$\textrm{SL}(3,\mathbb{R})$.
