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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

Your question is related to an open problem about finiteness of the index of an "arithmetic" group in its Zariski closure (see, for example, Peter Sarnak's lecture Thin Groups and the Affine Sieve as well as arXiv:math/0605675 and arXiv:0803.3322).

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