# Hausdorff dimension of products of normal subgroups

Let $G$ be a metric group, and let $h$ be the associated Hausdorff dimension function on subsets of $G$. (See for instance Barnea and Shalev, Hausdorff dimension, pro-p groups and Kac-Moody algebras, Trans. AMS 1997.) When do we have $h(AB) = h(A) + h(B) - h(A \cap B)$ for normal subgroups of $G$? If this property fails, is there still a general way to use $h$ (or some similar 'dimension' function) to construct a pseudometric on the lattice of normal subgroups of $G$? What I am looking for here are some informative examples of bad behaviour, particularly for profinite groups.

-

In the first conference I ever went to Slava Grigorchuk asked me a similar question and I didn’t have an answer. But when I have got back to Jerusalem I have talked with Elon Lindenstrauss about it and he suggested the following easy counterexample. Take $G=\mathbb{F}_p[[t]]$. Pick $S$ to be a subset of the integers with density one and with infinite complement $T$. Say $S=\left\{ n_i \right\}$ and $T=\left\{ m_i \right\}$. Take $A=\overline{< t^{n_i}>}$ and take $B=\overline{\left< t^{n_i} +t^{m_i} \right>}$. Cleary, $AB=G$, $h(A)=h(B)=1$, but $A \cap B=\emptyset$.
Now, $G$ is not finitely generated, if you would like to have a counterexample which is finitely generated, then you can take $G=SL_d(F_p[[t]])$ and construct in a similar way to the above $A$ which is made from upper triangular matrices and $B$ which is made from lower triangular matrices. However, $A$ and $B$ will not be normal any more.
I am not familiar with a counterexample in which $A$ and $B$ are normal and $G$ is finitely generated. I am also not familiar with a counterexample in which $A$ and $B$ are finitely generated. But as you can deduce from my story above this does not mean much.
Thanks, those are useful examples. It would also be interesting to find an example where $h(AB) > h(A) + h(B) - h(A \cap B)$, if this can occur. –  Colin Reid Sep 3 '10 at 7:27
Andrei Jaikin used Huasdorff dimension when he studied word width in $p$-adic analytic groups. It was not for subgroups but subsets in general. But you might find some ideas there that could be useful. –  Yiftach Barnea Sep 3 '10 at 7:49