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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.

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Is this question well-posed as it stands? The embedding per Q2 controls much more of the geometry, it seems, than the abstract Cayley graph, which may just be some tree (for a free group). For example at en.wikipedia.org/wiki/Fuchsian_group#Limit_sets you can read about a large class of Cantor sets, but these are associated to a definite group action, not an abstract group. –  Charles Matthews Jun 1 '10 at 21:02
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The Cayley graph of \mathbb{F}_2 is quite easy to describe! The Cayley graph of F_2 is another matter entirely... –  Qiaochu Yuan Jun 1 '10 at 21:27
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2 Answers 2

The Cayley graph of a finitely generated countable group is a metric space with respect to the word metric. One includes both edges and vertices in the geometric realization (after all, it's called a graph, not a discrete point set). If the group is nontrivial, the graph is regular of constant finite valence, so small neighborhoods of any point are isometric to line segments or stars. If the group is nontrivial, the Hausdorff dimension of the graph is 1, and this is independent of the choice of generating set. Any countable Cayley graph can be embedded (non-isometrically) in $\mathbb{R}^3$ using an enumeration of vertices by natural numbers, e.g., sending the $n$th vertex to $(n,n^2,n^3)$, and taking edges to be straight lines.

The free group on 2 letters can be embedded (non-isometrically) in $\mathbb{R}^2$ as a fractal, as shown in the linked image you gave. The Hausdorff dimension of the image in general strongly depends on the choice of embedding. For example, the Cayley graph of $\mathbb{Z}$ with generator $1$ is isometric to the real line, but you may be able to embed it in a larger space using some kind of Brownian motion, where the image will have Hausdorff dimension 2 almost surely.

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The key feature of the free group example is that the ends of the Cayley graph form a Cantor set. Charles mentioned in the comments that interesting fractal sets arise as limit sets of Fuchsian groups (the group acts on a hyperbolic space and the limit set lives in its boundary, which may be modeled by a sphere). A breathtaking pandemonium of pictures appears in "Indra's Pearls" by Mumford, Series, and Wright (the mathematical text is excellent as well).

For abstract finitely generated groups, one looks for a "boundary" that is an appropriate replacement of the limit set. For hyperbolic groups, there is a natural notion of Gromov boundary which can be constructed from the Cayley graph and does not depend on the choice of the finite generating set. Cantor set and its higher-dimensional analogues, such as the Menger sponge, are known to arise in this way. Here is a nice survey.

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