REPRESENTATIONS OF SURFACE GROUPS IN THE PROJECTIVE GENERAL LINEAR GROUP

Abstract

Given a closed, oriented surface X of genus g >= 2, and a semisimple Lie group G, let R-G be the moduli space of reductive representations of pi(1) X in G. We determine the number of connected components of R-PGL(n,R- R), for n >= 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in R-SL(3,R-R) is homotopically equivalent to R-SO(3).