: For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: 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 form a single quasi-isometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.

In the past decades, there has been considerable progress in the theory of random walks on groups acting on hyperbolic spaces. Despite the abundance of such groups, this theory is inherently not preserved under quasi-isometry. In this talk, I will present our study of random walks on groups that satisfy a certain QI-invariant property that does not refer to an action on hyperbolic spaces. Joint work with Kunal Chawla, Kasra Rafi, and Vivian He.