Differentiable conjugacy for groups of area preserving circle diffeomorphisms

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