New York Journal of Mathematics
Volume 21 (2015) 383-416


Sean Lawton and Daniel Ramras

Covering spaces of character varieties

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Published: June 16, 2015
Keywords: character variety, moduli space, covering, path components, fundamental group, universal cover, properly discontinuous action
Subject: 57M10, 14D20, 32G13, 14L30

Let Γ be a finitely generated discrete group. Given a covering map H→ G of Lie groups with G either compact or complex reductive, there is an induced covering map Hom(Γ, H)→ Hom(Γ, G). We show that when π1(G) is torsion-free and Γ is free, free Abelian, or the fundamental group of a closed Riemann surface Mg, 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. As an application, we show that for g≧1, the stable moduli space Hom(π1Mg, SU)/SU is homotopy equivalent to CP.

In the Appendix, Ho and Liu show π0Hom(π1Mg, G) and π1[G,G] are in bijective correspondence for all complex connected reductive Lie groups G.


The first author was partially supported by the Simons Foundation (#245642) and the NSF-DMS (#1309376). The second author was partially supported by the Simons Foundation (#279007).

Author information

Sean Lawton:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, Virginia 22030, USA

Daniel Ramras:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford, LD 270, Indianapolis, IN 46202, USA