香蕉视频

Title: The thorny search for a spine.

Abstract: A spine for a group G acting properly discontinuously on a space E is a subset onto which there is a G-equivariant deformation retraction of E. For the space of lattices of covolume 1 in R^n, the action of SL_n(Z) admits a spine of minimal dimension called the well-rounded retract, consisting of the lattices whose shortest nonzero vectors span R^n. Whether an analogous spine of dimension 4g-5 exists for the action of the mapping class group on the Teichmuller space of closed hyperbolic surfaces of genus g is an open problem. In a 1985 preprint, Thurston claimed to prove that the set X_g of surfaces of genus g whose systoles (the shortest closed geodesics) fill (cut the surfaces into polygons) is a spine for the mapping class group. However, his argument had a serious gap. Whether or not X_g is a spine, I will explain why its dimension is strictly larger than 4g-5 in certain genera. The same construction shows that the set of surfaces whose systoles generate a finite-index subgroup in homology (a closer analogue of the well-rounded retract) does not contain any spine.