Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface and related surfaces

TL;DR

This paper provides a new geometric description of the genus 3 Arnoux-Yoccoz surface using Delaunay polygons and classifies it within families of surfaces with specific symmetry groups.

Contribution

It introduces a novel polygonal representation of the Arnoux-Yoccoz surface and situates it within families characterized by dihedral symmetries.

Findings

The Arnoux-Yoccoz surface can be described via Delaunay polygons.

It belongs to two families of surfaces with dihedral symmetry.

The isometry groups include the dihedral group of the square.

Abstract

We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.