Flexibility of surface groups in classical simple 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. García-Prada and P. Gothen.