The genus of Cayley graphs on triangular billiard surfaces

Grzegrzolka, Joanna; McCartney, Jaime Lynne; Schmurr, Jason

doi:10.2140/involve.2025.18.567

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The genus of Cayley graphs on triangular billiard surfaces

Joanna Grzegrzolka, Jaime Lynne McCartney and Jason Schmurr

Vol. 18 (2025), No. 4, 567–582

DOI: 10.2140/involve.2025.18.567

Abstract

We discuss a connection between two well-known constructions in mathematics: Cayley graphs and rational billiard surfaces. We describe a natural way to draw a Cayley graph of a dihedral group on each rational billiard 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 a Cayley graph associated with a triangular billiard table is always 0 or 1. One reason this is interesting is that there exist triangular billiard surfaces of arbitrarily high genus, so the genus of the associated graph is often much lower than the genus of the billiard surface.

Keywords

rational billiards, graph genus, Cayley graphs, low-dimensional topology

Mathematical Subject Classification

Primary: 05C10, 37C83

Milestones

Received: 28 June 2022

Revised: 28 March 2024

Accepted: 1 April 2024

Published: 30 July 2025

Communicated by Ann N. Trenk

Authors

Joanna Grzegrzolka
Lee University Cleveland, TN United States

Jaime Lynne McCartney
Dalton State College Dalton, GA United States

Jason Schmurr
Lee University Cleveland, TN United States

Vol. 18, No. 4, 2025