Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Gl\"ockner's inverse function theorem, we show that the linear action of a compact Lie group on a Fr\'echet space admits a slice. Second, using the Nash--Moser theorem, we establish a slice theorem for the tame action of a tame Fr\'echet Lie group on a tame Fr\'echet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification

Relative Property (T) Actions and Trivial Outer Automorphism Groups

We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in particular the first examples of such actions for the free group on $2$ generators.Comment: Improvement in the presentation of the replacement trick. Introduction of compressions for non-ergodic equivalence relation