The topology and geometry of random square-tiled surfaces

Abstract: A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk we will consider a randomizing model for STSs based on permutation pairs and use it to compute the genus distribution. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. In this talk, we obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

dynamical systems

Audience: researchers in the topic

( video )

Comments: Zoom link: unipd.zoom.us/j/92074073847?pwd=ZDd6Y28yUVowcjJNUFFRYU52WGtTdz09

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