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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中文翻译：

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马鞍连接图的大规模几何

摘要：我们证明了与任何半平移曲面相关的鞍形连接图是 $4$-双曲且与正则可数无穷价树一致拟等距。因此，鞍座连接图不是准等距刚性的。我们还将其 Gromov 边界描述为一组没有鞍状连接的直叶。在我们的论点中，我们给出了可能具有独立兴趣的弧图中独角兽路径的概括。

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