A series of coverings of the regular n-gon

TL;DR

This paper constructs infinite series of translation coverings of regular n-gons, revealing new examples of infinite translation surfaces with lattice Veech groups for both odd and even n.

Contribution

It introduces explicit infinite series of translation coverings of regular n-gons with shared or constant Veech groups, expanding the understanding of lattice Veech groups in translation surfaces.

Findings

Constructed infinite series of coverings for odd n ≥ 5

Constructed infinite series of coverings for even n ≥ 8

Provided explicit examples of infinite translation surfaces with lattice Veech groups

Abstract

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to explicit examples of infinite translation surfaces with lattice Veech group.