In this paper, we construct spines, i.e., -equivariant deformation retracts, of the Teichmüller space of compact Riemann surfaces of genus . Specifically, we define a -stable subspace of positive codimension and construct an intrinsic -equivariant deformation retraction from to . As an essential part of the proof, we construct a canonical -deformation retraction of the Teichmüller space to its thick part when is sufficiently small. These equivariant deformation retracts of give cocompact models of the universal space for proper actions of the mapping class group . These deformation retractions of are motivated by the well-rounded deformation retraction of the space of lattices in . We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichmüller space.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