Combinatorial complexes for translation surfaces and dynamics on moduli space

大部分的研究一直致力于理解与平移表面相关的组合复合物。主要结果涉及鞍形连接复合体的几何和组合结构，它编码的平移表面上的直线段的相交模式。我们与我的合作者安雅·兰德克（Anja Randecker）和瓦伦蒂娜·迪萨罗（Valentina Disarlo）建立了鞍形连接复合体的刚性：组合结构完全控制了相关平移曲面的仿射几何。因此，平移曲面上的直线轨迹的动力学性质也被编码在鞍形连接复形的组合学中。我还与Randecker，Disarlo，and Huiping Pan一起对鞍形连接复形的大尺度几何进行了进一步的研究。我们已经确定了鞍形联络复形的Gromov边界与不包含鞍形联络的直线叶理空间。我们还证明了每一个鞍联络复形都是正则无限叶树的拟等距，因此它们都属于一个拟等距类。这表明鞍形联络复形的精细几何和粗糙几何表现出截然不同的行为。另一个结果是关于平移曲面的仿射对称性的结果。我已经证明，任何稳定Teichmueller盘的映射类群的子群都是不失真的。因此，任何Veech曲面的仿射仿射群都是不失真的。

Much of the research has been dedicated to understanding combinatorial complexes associated to translation surfaces. The main results relate to the geometry and combinatorial structure of the saddle connection complex, which encodes the intersection pattern of straight-line segments on a translation surface. With my collaborators Anja Randecker and Valentina Disarlo, we established rigidity of the saddle connection complex: the combinatorial structure completely governs the affine geometry of the associated translation surface. Consequently, the dynamical properties of straight-line trajectories on a translation surface are also encoded in the combinatorics of the saddle connection complex.I am also conducting further research on the large-scale geometry of the saddle connection complex with Randecker, Disarlo, and Huiping Pan. We have identified the Gromov boundary of the saddle connection complex with the space of straight-line foliations that contain no saddle connections. We have also shown that every saddle connection complex is quasi-isometric to the regular infinite-valent tree, and so they all belong to a single quasi-isometry class. This demonstrates that the fine and coarse geometry of the saddle connection complex exhibit very contrasting behaviours.Another outcome is a result regarding affine symmetries of translation surfaces. I have shown that any finitely generated subgroup of the mapping class group that stabilises a Teichmueller disc is undistorted. Consequently, the affine diffeomorphism group of any Veech surface is undistorted.