  For $s >\frac{3}{2}$, the group of Sobolev class s diffeomorphisms of the
circle is a smooth manifold modeled on the space of Sobolev class s sections of
the tangent bundle of the circle. It is a topological group in the sense that
multiplication given by the composition of applications is well-defined and
continuous, the inverse is continuous, left translation is continuous and right
translation is smooth. These results are consequences of the Sobolev Lemma. For
the same reasons, the subgroup of Sobolev class s diffeomorphisms of the circle
preserving three points, is, for $s >\frac{3}{2}$ a smooth manifold and a
topological group modeled on the space of Sobolev class s vector fields
vanishing at these three points. One may ask what happens for the critical
value $\frac{3}{2}$ and look for a group with some regularity and a manifold
structure such that the tangent space at the identity is isomorphic to the
space Sobolev class $\frac{3}{2}$ vector fields vanishing at three given
points. The universal Teichm\"uller space verify these conditions.
