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. $S{O}^{\ast }(2n)$, $n$ odd) and the surface group is maximal in some $S(U(p,p)\times U(q-p))\subset SU(p,q)$ (resp. $S{O}^{\ast }(2n-2)\times SO(2)\subset S{O}^{\ast }(2n)$). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.