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deformation retraction of the space of lattices in n. We also include a summary of results and difficulties of an unpublished paper of Thurston on a potential spine of the Teichm\"uller space.