Representations of surface groups in the projective general linear group

TL;DR

This paper classifies the connected components of the moduli space of reductive representations of surface groups into PGL(n,R) for even n≥4, using Higgs bundle theory and topology of real projective bundles.

Contribution

It provides a complete classification of connected components of R_PGL(n,R) for even n≥4 and relates the topology of certain representation spaces to known geometric structures.

Findings

Number of connected components of R_PGL(n,R) for even n≥4 determined.

Real projective bundles over surfaces classified for the first time.

Homotopy equivalence established between the complement of the Hitchin component in R_SL(3,R) and R_SO(3).

Abstract

Given a closed, oriented surface X of genus g>1, and a semisimple Lie group G, let R_G be the moduli space of reductive representations of the fundamental group of X in G. We determine the number of connected components of R_PGL(n,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) is homotopically equivalent to R_SO(3).