Nonhomogeneous locally free actions of the affine group

Abstract

We classify locally free smooth actions of the real affine group on closed three-dimensional manifolds up to smooth conjugacy. In particular, we show that there are non-homogeneous actions when the manifold is not a rational homology sphere and its fundamental group is not solvable. We say two right actions ρ1 and ρ2 of a Lie group G on manifolds M1 and M2 are C ∞ conjugate if there exist an isomorphism σ of G and a C ∞ diffeomorphism H from M1 to M2 such that H(ρ1(p, g)) = ρ2(H(p), σ(g)) for any g 2 G and p 2 M1. The map H is called a C ∞ conjugacy between ρ1 and ρ2. We also say ρ1 and ρ2 are C ∞ orbit equivalent if there exists a C ∞