Finite translation surfaces with maximal number of translations

TL;DR

This paper investigates finite translation surfaces with the largest possible automorphism group size, called Hurwitz translation surfaces, and explores their existence across different genera.

Contribution

It characterizes the existence of Hurwitz translation surfaces of genus g, extending classical concepts to translation surfaces.

Findings

Maximal automorphism groups are bounded by genus g

Existence of Hurwitz translation surfaces depends on genus g

Provides classification criteria for Hurwitz translation surfaces

Abstract

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maximal bound Hurwitz translation surfaces. We study for which g there exist Hurwitz translation surfaces of genus g.