Property FW, differentiable structures, and smoothability of singular actions

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.

arXiv comment20 pages, to appear in Journal of Topology

References in corpus (12)

• Property (T) and rigidity for actions on Banach spaces
• Extensions of amenable groups by recurrent groupoids
• Group actions with commensurated subsets, wallings and cubings
• Free groups of interval exchange transformations are rare
• Euler characteristics and actions of automorphism groups of free groups
• Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifolds
• Irreducible lattices, invariant means, and commensurating actions
• Commensurating actions of birational groups and groups of pseudo-automorphisms
• Symmetries of flat manifolds, Jordan property and the general Zimmer program
• A finitely presented infinite simple group of homeomorphisms of the circle
• Hyperbolicity as an obstruction to smoothability for one-dimensional actions
• $C^1$ actions on manifolds by lattices in Lie groups

Cited by in corpus (1)

• Commensurating actions for groups of piecewise continuous transformations