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

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

[学科分类] 神经科学

[关键词] Curve complex;boundary at infinity;relative Teichmüller space;measured foliation