Well-rounded equivariant deformation retracts of Teichmüller spaces

Abstract

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 We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.