Property FW, differentiable structures, and smoothability of singular actions

TL;DR

This paper establishes a criterion for smoothing group actions on manifolds with singularities, showing that under certain properties, such actions can be conjugated to smooth actions on possibly different manifolds, extending key results in the field.

Contribution

It introduces a smoothening criterion for singular diffeomorphism actions and generalizes existing results to actions with countably many singularities.

Findings

Aperiodic actions with singularities can be conjugated to smooth actions.

Results extend Navas's theorem and Zimmer's conjecture solutions to singular actions.

Group properties like FW and (T) are crucial for smoothability.

Abstract

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.