This is related to the question $G=\langle a\rangle H$ for subgroup $H$ raised a few days ago. Suppose $\Gamma $ is a higher rank lattice (for example, $SL_3({\mathbb Z})$). As Misha says in his comments, by this, I mean an irreducible lattice in a linear semi-simple Lie group $G$ of real rank at least two such that $G$ has no compact factors. Can one find a subgroup $H$ of infinite index in $\Gamma$ and an infinite cyclic subgroup $A$ of $\Gamma$ such that $\Gamma$ is the set theoretic product $AH$?

When I was a graduate student, this question was "floating around"; I believe the question is due to Robert Zimmer.