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 M^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. As an application, we show that for g≧1, the stable moduli space Hom(π₁M^g, SU)/SU is homotopy equivalent to CP^∞.

In the Appendix, Ho and Liu show π₀Hom(π₁M^g, 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).