For a half-translation surface, its saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. It can be seen as a subgraph of the arc graph of the underlying topological surface. In this talk, I will discuss both the fine and large-scale geometry of the saddle connection graph. We show that any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasi-isometry type: all saddle connection graphs are quasi-isometric. We will also discuss a characterization of its Gromov boundary.
This is based on joint work with Robert Tang, Huiping Pan, and Anja Randecker.