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][1]. 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 $G$, that we still call $G$, 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][2]...

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Another random thought was originated by an answer to [this][3] 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?



  [1]: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
  [2]: http://en.wikipedia.org/wiki/Horseshoe_map
  [3]: http://mathoverflow.net/questions/38806/proper-subgroup-of-gln-z-isomorphic-to-gln-z