Strongly dense representations of surface groups

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 $\textrm{SL}(3,\mathbb{R})$ .