Cayley Graphs on Billiard Surfaces, and Their Genus

TL;DR

This paper explores the relationship between Cayley graphs and rational billiard surfaces, proving that for triangular billiard surfaces, the associated Cayley graph's genus is always zero or one, regardless of the surface's genus.

Contribution

It establishes a bound on the genus of Cayley graphs derived from triangular billiard surfaces, showing they are always low-genus graphs.

Findings

Cayley graph genus is at most one for triangular billiard surfaces.

Billiard surfaces can have arbitrarily high genus, but their associated Cayley graphs do not.

The result links geometric properties of billiard surfaces to combinatorial graph invariants.

Abstract

In this article we discuss a connection between two famous constructions in mathematics: a Cayley graph of a group and a (rational) billiard surface. For each rational billiard surface, there is a natural way to draw a Cayley graph of a dihedral group on that surface. Both of these objects have the concept of "genus" attached to them. For the Cayley graph, the genus is defined to be the lowest genus amongst all surfaces that the graph can be drawn on without edge crossings. We prove that the genus of the Cayley graph associated to a billiard surface arising from a triangular billiard table is always zero or one. One reason this is interesting is that there exist triangular billiard surfaces of arbitrarily high genus , so the genus of the associated graph is usually much lower than the genus of the billiard surface.