Maximal representations of surface groups in bounded symmetric domains

Abstract:

Let $\Gamma$ be the fundamental group of a hyperbolic surface of genus $g$; for $1 \le p \le q,PSU(p,q)$ is the group of isometries of a certain Hermitian symmetric space ${D_{p,q}}$ of rank $p$. There exists a characteristic number $c:\operatorname {Hom} (\Gamma ,PSU(p,q)) \to \mathbb {R}$, which is constant on each connected component and such that $|c(\rho )| \leq 4p\pi (g - 1)$ for every representation $\rho$. We show that representations with maximal characteristic number (plus some nondegeneracy condition if $p > 2$ leave invariant a totally geodesic subspace of ${D_{p,q}}$ isometric to ${D_{p,p}}$.