## Cayley digraphs and fractals

I have been thinking recently about the generation graph of a finite group: this is the graph whose vertices are the group elements, two vertices joined by an edge if together they generate the group. Of course the graph is not very interesting unless the group can actually be generated by two elements …

Talking to the summer research students today, an interesting thought struck me. Let G be a group which can be generated by two elements. Then there is a natural connection between the generation graph for G and the Cayley digraphs for G with out-degree 2:

A pair x,y of group elements form an edge in the generation graph if and only if the Cayley digraph Cay(G,{x,y}) is connected.

Does this remind you of anything? In the words of Wikipedia,

A point is in the Mandelbrot set exactly when the corresponding Julia set is connected.

So the relation between edges of the generation graph and divalent Cayley digraphs is “the same” as the relationship between the Mandelbrot set and Julia sets.

Is there any more to it?

Incidentally, as you would expect, the Wikipedia article has some nice pictures!