Affine equivalence and saddle connection graphs of half-translation surfaces

TL;DR

This paper studies the structure of saddle connection graphs on half-translation surfaces, proving that their automorphisms correspond to affine homeomorphisms, and explores the automorphism groups and quotient graphs.

Contribution

It establishes a correspondence between graph isomorphisms and affine homeomorphisms for half-translation surfaces, and analyzes the automorphism groups of these graphs.

Findings

Isomorphisms of saddle connection graphs are induced by affine homeomorphisms.

Automorphism groups of saddle connection graphs are characterized.

Quotient graphs of saddle connection graphs are studied.

Abstract

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the saddle connection graph, and the corresponding quotient graph.