Zariski dense discontinuous surface groups for reductive symmetric spaces

Abstract

Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $\Gamma$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $\Gamma$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $\Gamma$ of high genus contained in a reductive group $G$, if the Zariski closure of $\Gamma$ is locally isomorphic to $SL(2,\mathbb{R})$, then $\Gamma$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$.