Question

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.

[Edit] 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.

Answer 1

The result to which you refer is not a result but a conjecture of A. Lubotzky. Long and Reid have constructed some examples. -- the relevant preprints can be found on Alan Reid's web page. I assume that Lubotzky's conjecture is about three and not two generators because he did not want to be too ambitious -- nobody knows anything concrete, to the best of my knowledge.