  We study groups of circle diffeomorphisms whose action on the cylinder
$\mathcal C=\mathbb S^1\times \mathbb S^1\setminus \Delta$ preserves a volume
form. We first show that such a group is topologically conjugate to a subgroup
of $\rm{PSL}(2,\mathbb R)$, then discuss the existence of a differentiable
conjugacy.
