Density of Zariski density for surface groups

TL;DR

This paper demonstrates that surface groups in reductive real algebraic groups can typically be deformed to become Zariski dense, except in specific symmetric space cases, providing a partial converse to known rigidity results.

Contribution

It establishes a deformation result for surface groups into reductive groups, identifying conditions under which they become Zariski dense, extending understanding of group representations.

Findings

Surface groups can be deformed to Zariski dense groups.

Exceptions occur when the Zariski closure acts transitively on certain symmetric spaces.

The result complements existing rigidity theorems by Burger, Iozzi, and Wienhard.

Abstract

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.