Large-scale geometry of the saddle connection graph

2020/11/25 by Valentina Disarlo, Disarlo, Valentina, Huiping Pan +5 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2011.12975

openalex publication_date 2020/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Geometry of the mapping class groups III: Quasi-isometric rigidity
• Cylinder curves in finite holonomy flat metrics

Cited by

• Epimorphism classes and relatively maximal metrics in large-scale geometry

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