Translation Covers Among Triangular Billiard Surfaces

TL;DR

This paper classifies all translation covers among triangular billiard surfaces using holonomy fields and a new geometric invariant called the fingerprint, advancing understanding of their structural relationships.

Contribution

It introduces the fingerprint invariant and applies holonomy field techniques to completely classify translation covers among triangular billiard surfaces.

Findings

Complete classification of translation covers among triangular billiard surfaces.

Introduction of the fingerprint as a geometric invariant.

Application of holonomy field methods to billiard surface analysis.

Abstract

We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.