Complete periodicity of Prym eigenforms

TL;DR

This paper investigates the directional flow on Prym eigenforms, proving algebraic and complete periodicity properties, and explores the structure of their Veech groups, including new examples with infinitely generated groups.

Contribution

It establishes algebraic and complete periodicity for Prym eigenforms and constructs new translation surfaces with complex Veech groups.

Findings

Homological directions are algebraically periodic.

Directions of regular closed geodesics are completely periodic.

Veech groups can be infinitely generated of the first kind.

Abstract

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation surfaces satisfying the topological Veech dichotomy. As a corollary we obtain new translation surfaces whose Veech group is infinitely generated and of the first kind.