I was wondering what has been done concerning the Hausdorff measure of the Cayley graphs of finitely generated countable groups. There are number of issues that would need to be dealt with:

1.) By the Cayley graph should we mean just the vertices or the entire 1-skeleton?

2.) How could one embed such a graph into $\mathbb{R}^n$?

3.) How does the Hausdorff dimension change with a change of generating set?

I nice example of somewhere to start would be the Cayley graph of $\mathbb{F}_2$, the free group on 2 generators, with the two generators as the generating set.

A nice picture of this, with a possible embedding into Euclidean space can be found at http://en.wikipedia.org/wiki/File:Cayley_graph_of_F2.svg

Any comments on this would be very much appreciated.