Large-scale geometry of the saddle connection graph

TL;DR

This paper proves that the saddle connection graph of any half-translation surface is 4-hyperbolic, quasi-isometric to an infinite-valent tree, and characterizes its Gromov boundary, revealing its geometric properties and boundary structure.

Contribution

It establishes the hyperbolicity and quasi-isometric properties of the saddle connection graph and generalizes unicorn paths, providing new insights into its geometric structure.

Findings

The saddle connection graph is 4-hyperbolic.

It is quasi-isometric to an infinite-valent tree.

The Gromov boundary corresponds to straight foliations without saddle connections.

Abstract

We prove that the saddle connection graph associated to any half-translation surface is 4-hyperbolic and uniformly quasi-isometric to the regular countably infinite-valent tree. Consequently, the saddle connection graph is not quasi-isometrically rigid. We also characterise its Gromov boundary as the set of straight foliations with no saddle connections. In our arguments, we give a generalisation of the unicorn paths in the arc graph which may be of independent interest.