Zariski dense discontinuous surface groups for reductive symmetric spaces

TL;DR

This paper demonstrates the existence of Zariski-dense surface groups with high genus that can be deformed within reductive symmetric spaces, revealing new structures of discontinuous groups and their deformations.

Contribution

It establishes conditions under which surface groups in reductive symmetric spaces admit small deformations and identifies Zariski-dense surface subgroups acting properly discontinuously.

Findings

High genus surface groups can be deformed within reductive groups.

Existence of Zariski-dense surface subgroups acting properly discontinuously.

Deformations preserve the real rank of the Zariski closure.

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 $G$.