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My -possibly flawed- mental picture of free products of groups certainly comes from the special case usually performed to illustrate the construction that proves the Banach-Tarski paradox. Thus I'm used to think of a free product as a certain kind of self-similar "fractal".

Although from the set theoretical or algebraic viewpoint it may be clear, one may ask if this "fractal" property is also expressible in metric/topological terms, and if it is possessed by free products of, say, discrete groups in general (not only $\mathbb{Z}*\mathbb{Z}$).

I would like to know if there is a more or less natural way to put a topology (or even a metric) on the free product $G*H$ of two discrete groups $G$ and $H$ so that the above "fractal" picture is retained.

I suppose the topology should satisfy:

  1. compatibility with group structure: $G*H$ should be a topological group;
  2. the subgroup of $G*H$ made of words with letters from $G$ (resp., from $H$), that we still call $G$ (resp., $H$), should inherit the discrete topology

In case we even look for a metric on $G*H$, I think it would be reasonable to require that

  • left (and right) translations should be homotheties: $d(gx,gy)=C\cdot d(x,y)$ for a constant $C=C(g)$ depending only on $g$,

and maybe:

  • $G*H$ should have fractionary Hausdorff dimension.

Perhaps a metric would be something like the metric one can put on the "space of words" used in the symbolic dynamics description (if I don't misremember - I'm no expert) of the horseshoe map...

Another random thought was originated by an answer to this MO question. It was noted that the modular group $PSL(2,\mathbb{Z})$ is isomorphic to the free product of cyclic groups $\mathbb{Z}/2*\mathbb{Z}/3$, and that it is not co-Hopf, i.e. it does have a proper subgroup isomorphic to itself (so, in a certain vein not dissimilar to the above one, it is "self similar").

May this fact have any interesting consequences about some properties of modular forms?

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It seems to me (but I'm no expert) that the self-similarity you mention is a property of the way the free group is acting on certain subsets of $\mathbb R^3$ rather than a property of the free group itself. – Yemon Choi Sep 20 '10 at 22:25
I also think that your second question deserves to be separated from this one. – Yemon Choi Sep 20 '10 at 22:26
I feel like you should really be looking at the Cayley graph of the group, which has the self-similarity you want for free groups as well as a natural metric (the word metric). – Qiaochu Yuan Sep 20 '10 at 23:16
Yemon, regarding your first comment, it's hard to know, as the OP hasn't been at all clear about what he means by the `fractal' property of the free group, but if he is talking about the fact that the free group is equidecomposable with itself, then this is indeed a property of the free group itself - another way of saying it is that the free group is non-amenable. If I remember correctly, the only really important fact about the action on $\mathbb{R}^3$ is that it's free on the unit sphere. – HJRW Sep 20 '10 at 23:55
It may helpful to think of this desired topology in terms of which sequences should converge. If $G$ and $H$ are to be discrete subgroups, then we need only address the question of convergence of sequences of elements $(x_n)$ where the length of $x_n$ tends to infinity. Here, by length, I mean the length of its normal form as an word of the from $x = a_1 a_2 \dots a_k$ where each $a_i$ is non-trivial and belongs alternately to either $G$ or $H$. Thus, you seem to be asking for a compactification of this group (and the fractional Hausdorf dimension will occur on its boundary). – Robert Bell Sep 21 '10 at 0:57

I wouldn't agree with your interpretation of free groups, I think that your understanding comes from the free group on 2 letters with a particular action on a fractal. That's not to say that there isn't much of interest to say about free products and actions on particular spaces.

But if you're just interested in free groups and how they are related to free products then I'd recommend reading up on Bass-Serre theory:

This studies how groups act on simplicial trees.

Current research is focused on the automorphisms of free groups, if you want to learn about this you should search for articles on "Outer space", possibly by Karen Vogtmann.

[EDIT: As pointed out the last paragraph points to just one topic of active research, which by no means describes fully the current research environment. ]

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+1 for your first three paragraphs. I would add to your last paragraph, which is a little categorical. Certainly, $\mathrm{Out}(F_n)$ is a very interesting and much-studied object, and outer space is an invaluable tool, but there are plenty of other aspects of free groups that are the subject of active research. To name a couple of examples close to my heart, the first-order theory of free groups and stable commutator length in free groups. Oh, and the Hanna Neumann Conjecture, of course. – HJRW Sep 21 '10 at 12:46
Whoops, thank you for pointing that out, very true. – James Griffin Sep 21 '10 at 15:01

The free product $\mathbf{Z} * \mathbf{Z}$ is typically viewed as a discrete group, since I believe that is the coproduct in the category of topological groups. Even if you topologize the free product by taking its image in $SO(3)$ under a paradoxical embedding, I think countability makes its Hausdorff dimension zero. I'm not sure what you mean by "fractal" geometry, but you can get something like a fractal from the Cayley graph (which has dimension one, unless you choose a non-uniform metric and take a completion).

In general, the free product of discrete groups has a discrete topology. If you choose generators of $G$ and $H$, you can construct a metric on the corresponding Cayley graph, and hence on the group, that is invariant under left translation (and gets shifted by at most constants under right translations). Again, the group has dimension zero, and the Cayley graph has dimension one. You can occasionally change the metric so that the completion of the graph and group acquire more dimension (e.g., by shrinking the ends so they have finite length), but this will destroy the homogeneity of the metric, even up to constants.

I don't think the particular group-theoretic property you mention says much about modular forms, because the abstract groups that are used do not have much intrinsic group-theoretic structure. For example, the kernel of reduction mod $N$ in $PSL_2(\mathbf{Z})$ is free for all $N>1$. What is more important is the particular way the groups act on the upper half plane, and on functions/sections of line bundles.

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