A Cayley Graph of PSL(2,7)

The smallest non-Abelian simple group is A5 or A₁(5) or A₁(4) (these are all isomorphic). It has 60 elements. Its Cayley graph can be drawn, without the arcs crossing, on a sphere: see four Cayley graphs of A5.

The second smallest non-Abelian simple group is PSL(2,7) (also known as A₁(7)) or GL(3,2) (also known as A₂(2)) these being isomorphic. It has 168 elements. Its Cayley graph cannot be drawn without the arcs crossing on a sphere, nor on a torus, nor on a genus-2 surface (like a torus but with two holes instead of one). It can be drawn on a surface of genus 3 (three holes). This page presents such a Cayley graph:

Explanation of pink parts of the diagram

To represent a genus 3 surface on your screen, the diagram uses "gluing instructions", in pink. You could in principle make a genus 3 surface, with this Cayley graph on it, as follows:

Print out the image above on a thin sheet of rubber.
Cut out and discard the pale pink regions.
Glue or sew together the cut edges in accordance with the pink single, double and triple arrows. The pink letters a-h, i-p, q-x provide further guidance on how things should join up.
To prevent the result from looking like a crumpled mess, pump it up with air, or stuff it with newspaper.

A more practicable way of making a version of this Cayley graph, nicely embedded in 3-space, is:

Take a cube of wood. Paint it white.
Drill three holes through it, each running between two opposite faces. Offset the holes so that they do not meet in the middle.
Copy the above diagram onto it, with the pink regions corresponding to the ends of the holes. It has been drawn so that it is clear what goes on each face. Some of the black and green lines will run through each hole.

Explanation of the Cayley graph itself

A Cayley graph uses a colour for each generator of the group, and its appearance depends fundamentally on the choice of generators. This Cayley graph uses two generators: one of order 2, shown by green arcs, and one of order 3, shown by black arcs. The black arcs need arrows on them to show which way they go; the green arcs go both ways and have no arrows.

The graph has obvious cubic symmetry. It has been drawn so as to emphasise one of the S4 subgroups of A₁(7). But there isn't really anything special about this subgroup: it is only one of seven conjugate S4s within A₁(7).

It is based on the tiling of the genus 3 surface with 24 heptagons. Each of these heptagons is identical, and has the same relationship to each of its seven neighbouring heptagons (though this symmetry is obscured in the diagram above, by the need to portray the structure on a flat screen).

Just as you can create a Cayley graph for A5 by tiling the sphere with 12 pentagons and then "truncating" the triangular vertices, you can do the same for the tiling of the genus-3 surface by 24 heptagons.

Some more Cayley diagrams
Some more pages on groups