JournalslemVol. 60, No. 1/2pp. 109–129

Well-rounded equivariant deformation retracts of Teichmüller spaces

  • Lizhen Ji

    University of Michigan, Ann Arbor, USA
Well-rounded equivariant deformation retracts of Teichmüller spaces cover

Abstract

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retraction of the space of lattices in Rn\mathbb R^n. We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.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