Representations of surface groups in the projective general linear group

Details Representations of surface groups in the projective general linear group Representations of surface groups in the projective general linear group

2009-01-15

arxiv.org/abs/0901.2314v2

Mathematics

Algebraic Geometry

47 pages; v2: various improvements and minor corrections suggested by the referee; to appear in Internat. J. Math

Scientific paper

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).

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