On visualization of the linearity problem for mapping class groups of surfaces

TL;DR

This paper investigates conditions under which mapping class groups of surfaces can be represented linearly, providing geometric and algebraic criteria for both punctured and closed surfaces.

Contribution

It introduces new linearity conditions for mapping class groups based on surface topology and group actions, advancing understanding of their linear representations.

Findings

Linearity conditions for once-punctured surfaces via deformation space actions

Linearity conditions for closed surfaces via isotopy classes of curves

Extension of conditions to surfaces with boundary, up to center

Abstract

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.