Property FW, differentiable structures, and smoothability of singular actions

We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group $\Gamma$ has the fixed point property FW for walls (e.g. if it has property (T)), every aperiodic action of $\Gamma$ by diffeomorphisms that are of class $C^r$ with countably many singularities is conjugate to an action by true diffeomorphisms of class $C^r$ on a homeomorphic (possibly non-diffeomorphic) manifold. As applications, we show that Navas's result for actions of Kazhdan groups on the circle, as well as the recent solutions to Zimmer's conjecture, generalise to aperiodic actions by diffeomorphisms with countably many singularities.