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We provide a complete classification of groups that can be realized as isometry groups of a translation surface M with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if S has no non-displaceable subsurfaces and its space of ends is self-similar, then every countable subgroup of GL+(2,R) can be realized as the Veech group of a translation surface X homeomorphic to M. The latter result generalizes and improves upon the previous findings of Przytycki-Valdez-Weitze-Schmithüsen and Maluendas-Valdez. To prove these results, we adapt ideas from the work of Aougab-Patel-Vlamis, which focused on hyperbolic surfaces, to translation surfaces.

Related project(s):
72Limits of invariants of translation surfaces

We determine the distribution of the number of saddle connections on a random translation surface of large genus. More specifically, for genus g tending to infinity, the number of saddle connections with lengths in a given interval [a/g, b/g] converges in distribution to a Poisson distributed random variable. Furthermore, the numbers of saddle connections associated to disjoint intervals of lengths are independent.

Related project(s):
72Limits of invariants of translation surfaces

We study finite abelian covers of the Chamanara surface, an example of a finite-area infinite translation surface with interesting dynamics and a large Veech group. Specifically, the Veech group of the Chamanara surface is a virtually free group on two generators. We characterize when finite abelian covers have large Veech groups themselves, namely when their Veech group has finite index in that of the Chamanara surface. For degree-2 covers, we provide a detailed analysis of these finite-index Veech groups. As an application, we prove that every free group arises as the projective Veech group of a finite-area infinite translation surface.

Related project(s):
72Limits of invariants of translation surfaces