Covering spaces of character varieties

Abstract: Let F be a finitely generated discrete group. Given a covering map H to G of Lie groups with G either compact or complex reductive, there is an induced covering map Hom(F, H) to Hom(F, G). We show that when the fundamental group of G is torsion-free and F is free, free Abelian, or the fundamental group of a closed Riemann surface M of genus g, this map induces a covering map between the corresponding moduli spaces of representations. We give conditions under which this map is actually the universal covering, leading to new information regarding fundamental groups of these moduli spaces. Let pi be the fundamental group of M. As an application, we show that for g>0, the stable moduli space Hom(pi, SU)/SU is homotopy equivalent to infinite complex projective space. In the Appendix by Ho and Liu, it is shown show that there is a bijection between the number of connected components of Hom(pi, G) and the fundamental group of [G,G] for all complex connected reductive Lie groups G.