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 $S$.