Cayley Graphs on Orientable 2-Manifolds

This is an index page to some pages showing some of the Cayley graphs that can be drawn on some orientable 2-manifolds. Each Cayley graph is drawn on the surface of the same genus as the group it portrays: so none of these Cayley graphs could have been drawn on a surface of lower genus.

• The plane or sphere. 26 finite Cayley graphs, 20 infinite. Many more exist.
• The torus. 27 finite Cayley graphs. Many more exist.
• The genus-2 manifold. Just one Cayley graph, for the only group of this genus.
• The genus-3 manifold. Just one Cayley graph, of the many that exist.

The diagrams for manifolds of genus higher than 1 use pink lines and labels to show how the manifolds are constructed from the diagrams. This is explained in the page Representation of 2-manifolds.