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

By E. Klarreich. Groups, Geometry, and Dynamics, 2018

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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 Teichm\"{u}ller space $\mathcal{T}_{el}(S)$ obtained from the Teichm\"{u}ller space by collapsing each region where a simple closed curve is short to be a set of diameter 1. $\mathcal{C}(S)$ and $\mathcal{T}_{el}(S)$ are quasi-isometric, and Masur-Minsky have shown that $\mathcal{C}(S)$ and $\mathcal{T}_{el}(S)$ are hyperbolic in the sense of Gromov. We show that the boundary at infinity of $\mathcal{C}(S)$ and $\mathcal{T}_{el}(S)$ is the space of topological equivalence classes of minimal foliations on $S$.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/33c94e1ad916d9f82b321c15012de359817bd014