Well-rounded equivariant deformation retracts of Teichm\"uller spaces

TL;DR

This paper constructs $ ext{Mod}_g$-equivariant deformation retracts of Teichm"uller space, providing cocompact models for the universal space of proper $ ext{Mod}_g$ actions, inspired by lattice space retractions.

Contribution

It introduces a new $ ext{Mod}_g$-stable subspace and constructs intrinsic deformation retractions of Teichm"uller space, advancing understanding of its geometric and group action properties.

Findings

Constructed a $ ext{Mod}_g$-equivariant deformation retraction to a subspace of positive codimension.

Developed a canonical deformation retraction to the thick part of Teichm"uller space.

Provided cocompact models for the universal space $ ext{E} ext{Mod}_g$ for proper actions.

Abstract

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichm\"uller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension and construct an intrinsic $\Mod_g$-equivariant deformation retraction from $\T_g$ to $S$. As an essential part of the proof, we construct a canonical $\Mod_g$-deformation retraction of the Teichm\"uller space $\T_g$ to its thick part $\T_g(\varepsilon)$ when $\varepsilon$ is sufficiently small. These equivariant deformation retracts of $\T_g$ give cocompact models of the universal space $\underline{E}\Mod_g$ for proper actions of the mapping class group $\Mod_g$. These deformation retractions of $\T_g$ are motivated by the well-rounded deformation retraction of the space of lattices in $\R^n$. We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichm\"uller space.