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?