Combinatorial complexes for translation surfaces and dynamics on moduli space

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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.