Train tracks and the Gromov boundary of the complex of curves

TL;DR

This paper provides a combinatorial proof linking the Gromov boundary of the complex of curves for a surface to the space of filling minimal geodesic laminations, clarifying a key boundary identification.

Contribution

It offers a combinatorial proof of Klarreich's result connecting the Gromov boundary with geodesic laminations, enhancing understanding of the complex's boundary structure.

Findings

Identifies the Gromov boundary with filling minimal geodesic laminations

Provides a combinatorial proof of Klarreich's unpublished result

Clarifies the topology of the boundary in the complex of curves

Abstract

We give a combinatorial proof of an unpublished result of E. Klarreich: The Gromov boundary of the complex of curves of a non-exceptional oriented surface S of finite type can naturally be identified with the space of minimal geodesic laminations on S which fill up S, equipped with a coarse Hausdorff topology.