Flexibility of surface groups in classical groups

TL;DR

This paper investigates the deformability of surface groups within classical Lie groups, showing they can generally be deformed to Zariski dense subgroups except in specific maximal cases related to certain subgroup structures.

Contribution

It provides a classification of when surface groups in classical Lie groups can be deformed to Zariski dense groups, extending rigidity results with new deformation criteria.

Findings

Surface groups in most classical groups can be deformed to Zariski dense groups.

Exceptions occur for specific maximal surface groups within certain subgroup configurations.

The results complement and extend existing rigidity theorems for surface groups in Lie groups.

Abstract

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is $SU(p,q)$ (resp. $SO^* (2n)$, $n$ odd) and the surface group is maximal in some $S(U(p,p)\times U(q-p))\subset SU(p,q)$ (resp. $SO^* (2n-2)\times SO(2)\subset SO^* (2n)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. Garc\'{\i}a-Prada and P. Gothen.