# Question related to the abelianization of simplectic groups

Let $H \subset \mathrm{Sp}(\mathbb{Z})$ be a subgroup of the simplectic group of (square of even dimension) matrices with integer entries, and let $H^{(\ell)}$ denote its pro-$\ell$ completion for some prime $\ell$. Then does $H$ being Zariski dense in $\mathrm{Sp}(\mathbb{Z})$ imply that the abelianization $(H^{(\ell)})^{ab}$ is finite?

Or does the converse hold (i.e. $(H^{\ell})^{ab}$ is finite for some $\ell$ implies $H < \mathrm{Sp}(\mathbb{Z})$ is Zariski dense)?

I have no idea how to go about solving this, and half expect that there are easy counterexamples that I'm not thinking of.

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The converse seems to be disproved by considering e.g. any finite subgroup of Sp$_n(\mathbb{Z})$. –  Tom Church Nov 10 '11 at 21:29
@Jeff: Could you please give some motivation for the question? –  Alain Valette Nov 11 '11 at 11:09