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

In this paper, we construct spines, i.e., ${\text{Mod}}_g$-equivariant deformation retracts, of the Teichm"uller space ${\text{T}}_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a ${\text{Mod}}_g$-stable subspace $S$ of positive codimension and construct an intrinsic ${\text{Mod}}_g$-equivariant deformation retraction from ${\text{T}}_g$ to $S$. As an essential part of the proof, we construct a canonical ${\text{Mod}}_g$-deformation retraction of the Teichm"uller space ${\text{T}}_g$ to its thick part ${\text{T}}_g(\epsilon )$ when $\epsilon$ is sufficiently small. These equivariant deformation retracts of ${\text{T}}_g$ give cocompact models of the universal space $\underset{‾}{E}{\text{Mod}}_g$ for proper actions of the mapping class group ${\text{Mod}}_g$. These deformation retractions of ${\text{T}}_g$ 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 We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichm"uller space.