Large-scale geometry of the saddle connection graph

Valentina Disarlo; Huiping Pan; Anja Randecker; Robert Tang

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arXiv · 2011.12975

Large-scale geometry of the saddle connection graph

Valentina Disarlo·Huiping Pan·Anja Randecker·Robert Tang

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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.

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Valentina Disarlo, Huiping Pan, Anja Randecker, Robert Tang. 2020-11-25. Large-scale geometry of the saddle connection graph. https://arxiv.org/abs/2011.12975

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