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In this article we discuss a connection between two famous constructions inmathematics: a Cayley graph of a group and a (rational) billiard surface. Foreach rational billiard surface, there is a natural way to draw a Cayley graphof 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 thelowest genus amongst all surfaces that the graph can be drawn on without edgecrossings. We prove that the genus of the Cayley graph associated to a billiardsurface arising from a triangular billiard table is always zero or one. Onereason this is interesting is that there exist triangular billiard surfaces ofarbitrarily high genus , so the genus of the associated graph is usually muchlower than the genus of the billiard surface.

Cayley Graphs on Billiard Surfaces, and Their Genus