Hyperbolicity as an obstruction to smoothability for one-dimensional actions

Ghys and Sergiescu proved in the 1980s that Thompson's group T, and hence F, admits actions by C-infinity diffeomorphisms of the circle. They proved that the standard actions of these groups are topologically conjugate to a group of C-infinity diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha and Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys and Sergiescu, we prove that the groups of Monod and Lodha and Moore are not topologically conjugate to a group of C-1 diffeomorphisms.