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.

27 pages; version 4 includes small modifications to Proposition 2.2 and Lemma 3.8, and corrects various typos

References in corpus (4)

• The topology of moduli spaces of free group representations
• Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups
• Fundamental Groups of Character Varieties: Surfaces and Tori
• The prequantum line bundle on the moduli space of flat $SU(N)$ connections on a Riemann surface and the homotopy of the large $N$ limit

Cited by in corpus (10)

• Homotopy Groups of Free Group Character Varieties
• Fundamental Groups of Character Varieties: Surfaces and Tori
• A survey on spaces of homomorphisms to Lie groups
• On spaces of commuting elements in Lie groups
• Principal Schottky Bundles over Riemann surfaces
• Flawed groups and the topology of character varieties
• On the second homotopy group of spaces of commuting elements in Lie groups
• The prequantum line bundle on the moduli space of flat $SU(N)$ connections on a Riemann surface and the homotopy of the large $N$ limit
• Bad Representations and Homotopy of Character Varieties
• When does the zero fiber of the moment map have rational singularities?