The boundary at infinity of the curve complex and the relative Teichmüller space

Abstract

In this paper we study the boundary at infinity of the curve complex \mathcal{C}(S) of a surface S of finite type and the relative Teichmüller space \mathcal{T}_{\mathrm{el}}(S) obtained from the Teichmüller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) are quasi-isometric, and Masur–Minsky have shown that \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) are hyperbolic in the sense of Gromov. We show that the boundary at infinity of \mathcal{C}(S) and \mathcal{T}_{\mathrm{el}}(S) is the space of topological equivalence classes of minimal foliations on S .

Topics & Concepts