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The genus of
Cayley graphs on triangular billiard surfaces
Joanna Grzegrzolka, Jaime Lynne McCartney and Jason Schmurr |
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Vol. 18 (2025), No. 4, 567–582
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Abstract |
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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. |
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Keywords
rational billiards, graph genus, Cayley graphs,
low-dimensional topology
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Mathematical Subject Classification
Primary: 05C10, 37C83
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Milestones
Received: 28 June 2022
Revised: 28 March 2024
Accepted: 1 April 2024
Published: 30 July 2025
Communicated by Ann N. Trenk
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Authors |
| Joanna Grzegrzolka | |
| Lee University Cleveland, TN United States |
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| Jaime Lynne McCartney | |
| Dalton State College Dalton, GA United States |
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| Jason Schmurr | |
| Lee University Cleveland, TN United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
