David Fisher (Lehman College)

Continuous and smooth orbit equivalence rigidity

Abstract. Let $\Gamma_1$ and $\Gamma_2$ be finitely generated groups and $\rho_1$ and $\rho_2$ actions of $\Gamma_1$ and $\Gamma_2$ on compact manifolds spaces M₁ and M₂. We call a C^k diffeomorphism $f:M_1{\rightarrow}M_2$ a C^k orbit equivalence if it maps orbits to orbits. We call $\rho_1$ C^k orbit equivalence rigid if any C^k orbit equivalence is equivariant.

We prove that the natural $SL_n(\mathbb Z)$ action on ${\mathbb T}^n$is C¹ orbit equivalence rigid and that the natrual ${\mathbb Z}^{n-1}$action on ${\mathbb T}^n$ is C⁰ orbit equivalence rigid,as well as many other results of a similar nature. For the C⁰ category there are results when M₁ is not a manifold, and we show that the orbit structure at infinity is a complete invariant for "most" hyperbolic groups. Unlike proofs of orbit equivalence rigidity in the measurable category, the proofs of our results are quite elementary and depend only on the countability of the group and the "size" of fixed point sets for group elements.

This is joint work with K. Whyte.