Action of the Johnson-Torelli group on representation varieties

Abstract

Let Σ be a compact orientable surface with genus g and n boundary components B = (B 1 , . . . , B n ). Let c = (c 1 , . . . , c n ) ∈ [−2, 2] n . Then the mapping class group MCG of Σ acts on the relative SU(2)-character variety X C := Hom C (π, SU(2))/SU (2), comprising conjugacy classes of representations ρ with tr(ρ(B i )) = c i . This action preserves a symplectic structure on the smooth part of X C , and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J ⊂ MCG be the subgroup generated by Dehn twists along null homologous simple loops in Σ. Then the action of J on X C is ergodic for almost all c.

Several Remarks on Groups of Automorphisms of Free Groups

“…The spaces of conjugacy classes (K × · · · × K)//K and actions of discrete groups on these spaces are widely discussed in theory of Teichmüller spaces and its neighborhood (on actions in L 2 , see [1], [2], [13]). …”