Dynamics on character varieties: a survey

Abstract. We survey recent work on the dynamics of the outer auto-morphism group Out(Γ) of a word hyperbolic group on spaces of (conju-gacy classes of) representations of Γ into a semi-simple Lie group G. All these results are motivated by the fact that the mapping class group of a closed surface acts properly discontinuously on the Teichmüller space of the surface. We explore two settings. 1) The work of Guichard, Labourie and Wienhard establishing the proper discontinuity of the action of outer automorphism groups on spaces of Anosov representations. 2) The work of Canary, Gelander, Lee, Magid, Minsky and Storm on the case where Γ is the fundamental group of a compact 3-manifold with boundary and G = PSL(2,C). 1. Overview It is a classical result, due to Fricke, that the mapping class group Mod(F) of a closed oriented surface F acts properly discontinuously on the Te-ichmüller space T (F) of marked hyperbolic (or conformal) structures on F. The mapping class group Mod(F) may be identified with an index two subgroup of the outer automorphism group Out(pi1(F)) of pi1(F) and Te-ichmüller space may be identified with a component of the spaceX(pi1(F),PSL(2,R)) of conjugacy classes of representations of pi1(F) into PSL(2,R). It is natural to attempt to generalize this beautiful result by studying the dynamics of the action of the outer automorphism group Out(Γ) of a finitely generated group Γ on the space X(Γ, G) of (conjugacy classes of) representations into a semi-simple Lie group G. We recall that Out(Γ) = Aut(Γ)/Inn(Γ) where Inn(Γ) is the group of inner automorphisms of Γ. We let X(Γ, G) = Hom(Γ, G)/G. In the case that G is a complex semi-simple Lie group, e.g. PSL(2,C), one may instead take the Mumford quotient (see, for example, Lubotzky-Magid [42]), which has the structure of an algebraic variety. This additional structure will not be relevant to our considerations. However, we will abuse notation in the traditional manner and refer to X(Γ, G) as the G-character variety of Γ