Large-scale geometry of the saddle connection graph
Geometric Topology
2020-12-10 v2 Combinatorics
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.
Keywords
Cite
@article{arxiv.2011.12975,
title = {Large-scale geometry of the saddle connection graph},
author = {Valentina Disarlo and Huiping Pan and Anja Randecker and Robert Tang},
journal= {arXiv preprint arXiv:2011.12975},
year = {2020}
}
Comments
28 pages, 9 figures; v2: corrected statement of Corollary 1.3 (not affecting any other part of the paper)