# 2 generated arithmetic groups

Suppose $G({\mathbb Z})$ is a higher rank non-cocompact arithmetic group (e.g. $SL_n({\mathbb Z})$ with $n\geq 3$, or $Sp_{2g}({\mathbb Z})$ with $g\geq 2$). I have seen a result (http://arxiv.org/abs/math/0409345) which says that every finite index subgroup $\Gamma$ of $G({\mathbb Z})$ contains a smaller finite index subgroup generated by three elements.

Does anyone know ANY example of $G({\mathbb Z})$, where three can be replaced by two? I believe Alan Reid has some result in this direction.

 That 2 should suffice is a conjecture, attributed to Alex Lubotzky. That $3$ DO suffice for non-uniform higher rank lattices in the result mentioned in the link. What I am asking is just ONE example where 2 generators suffice.

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Just a remark: for arithmetic subgroups of $PSL(2,C)$, it is conjectured there are finitely many 2-generated arithmetic groups. This is known to be a consequence of Lehmer's conjecture. – Ian Agol Mar 2 '13 at 16:57
Thank you. This is most interesting. – Venkataramana Mar 2 '13 at 17:03