Action of the Johnson-Torelli group on Representation Varieties

TL;DR

This paper studies how the Johnson-Torelli subgroup of the mapping class group acts on SU(2)-character varieties of surfaces, showing ergodic behavior in specific cases with implications for symplectic geometry and representation theory.

Contribution

It demonstrates the ergodic action of the Johnson-Torelli group on certain SU(2)-character varieties for genus one surfaces with two boundary components.

Findings

The action preserves a finite measure on the character variety.

Ergodicity holds for almost all boundary conditions c.

The study links surface topology with representation variety dynamics.

Abstract

Let \Sigma be a compact orientable surface with genus g and n boundary components B = (B_1,..., B_n). Let c = (c_1,...,c_n) in [-2,2]^n. Then the mapping class group MCG of \Sigma acts on the relative SU(2)-character variety X_c := Hom_C(\pi, SU(2))/SU(2), comprising conjugacy classes of representations \rho with tr(\rho(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 be the subgroup of MCG generated by Dehn twists along null homologous simple loops in \Sigma. Then the action of J on X_c is ergodic for almost all c.