gr.group theory - Hausdorff dimension of products of normal subgroups

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.

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