When is a group action determined by its orbit structure?

Abstract

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit equivalence is equivariant. Even in the presence of larger fixed sets, for actions preserving rigid geometric structures our results force sufficiently smooth orbit equivalences to be equivariant. For instance, if a countable group $Γ$ acts on ${\mathbb T}^n$ and the action is $C^1$ orbit equivalent to the standard action of $SL_n({\mathbb Z})$ on ${\mathbb T}^n$, then $Γ$ is isomorphic to $SL_n({\mathbb Z})$ and the actions are isomorphic. (The same result holds if we replace $SL_n(\mathbb Z)$ by a finite index subgroup.) We also show that preserving a geometric structure is an invariant of smooth orbit equivalence and give an application of our ideas to the theory of hyperbolic groups. In the course of proving our theorems, we generalize a theorem of Sierpinski which says that a connected Hausdorff compact topological space is not the disjoint union of countably many closed sets. We prove a stronger statement that allows "small" intersections provided the space is locally connected. This implies that for any continuous action of a countable group $\G$ on a connected, locally connected, locally compact, Hausdorff topological space, where the fixed set of every element is "small", every orbit equivalence is equivariant.