On train track splitting sequences

TL;DR

This paper proves that train track splitting sequences project to quasi-geodesics in the curve complex, introduces new concepts like induced tracks, and applies these results to show hyperbolicity of the disk complex and properties of train track graphs.

Contribution

It establishes the quasi-geodesic nature of train track splitting sequences in the curve complex and introduces new tools for analyzing train tracks.

Findings

Subsurface projection of splitting sequences is a quasi-geodesic.

Train track sliding and splitting sequences are quasi-geodesics in the train track graph.

Supports hyperbolicity of the disk complex.

Abstract

We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This result is an important step in the proof that the disk complex is Gromov hyperbolic. As another application we show that train track sliding and splitting sequences give quasi-geodesics in the train track graph, generalizing a result of Hamenstaedt [Invent. Math.].