An easy (horizontal) walk through fake octagons

TL;DR

This paper constructs an explicit infinite family of fake octagons, genus two translation surfaces with a single singular point, using simple cut-and-paste methods, and demonstrates their distinctness and approximability.

Contribution

It provides an elementary, explicit construction of infinitely many fake octagons, making the concept accessible and demonstrating their diversity and approximation properties.

Findings

Constructed an explicit infinite family of fake octagons.

Proved all iterates produce distinct fake octagons.

Showed any fake can be approximated by others in the family.

Abstract

A fake octagon is a genus two translation surface with only one singular point and the same periods as the octagon. Existence of infinitely many fakes was first established by McMullen in 2007, and more generally follows from dynamical properties of the so called isoperiodic foliation. The purpose of this note is to build an explicit infinite family of fake octagons, constructed by means of elementary methods. We describe an easy cut-and-paste surgery and show that the all iterates of that surgery produce fake octagons. We prove that any iterate gives a fake which is different from each other, and we show that any such fake can be approximated arbitrarily well by some other fake of the family. This note is intended to be elementary and fully accessible to non-expert readers.