Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits

TL;DR

This paper classifies finite orbits of the mapping class group acting on the moduli space of SL(2,C) representations of surface groups, identifying conditions for finite and bounded orbits across different genera.

Contribution

It provides a complete classification of finite and bounded orbits in the moduli space for surfaces of various genera, linking them to specific types of representations.

Findings

Finite orbits correspond to representations with finite image for genus ≥ 2.

Genus one finite orbits are associated with finite or special dihedral representations.

Results extend to bounded orbits in the moduli space.

Abstract

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.