# Well-rounded equivariant deformation retracts of Teichmüller spaces

### Lizhen Ji

University of Michigan, Ann Arbor, USA

## 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 $\mathcal T_g(\varepsilon)$ when $\varepsilon$ 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 We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.

## Cite this article

Lizhen Ji, Well-rounded equivariant deformation retracts of Teichmüller spaces. Enseign. Math. 60 (2014), no. 1, pp. 109–129

DOI 10.4171/LEM/60-1/2-6