Nonhomogeneous locally free actions of the affine group

Abstract

We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a nonhomogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.

Regular projectively Anosov flows on three-dimensional manifolds

“…We say an Anosov flow Φ is algebraic if there exists a Lie group G, its cocompact lattice Γ, and a one-parameter subgroup (1) It is different from but equivalent to the common definition of an Anosov flow as pointed out by Doering [12,Proposition 1.1]. (2) A PA flow with a smooth PA splitting is called regular. However, we do not use the term 'regular' in this sense since we use this term in other context below.…”

“…Applying Theorem 1.5 to O ρ , we obtain a classification of the orbit foliation of actions without the assumption on an invariant volume. In the forthcoming paper [2], we will give a classification of the actions of GA up to smooth conjugacy.…”

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

“…The proof is a combination of the following three steps. (1) Vanishing of cohomology ⇒ parameter rigidity. This is the sufficient condition for parameter rigidity proved in [17].…”

“…So we can use the rigidity theorems of Pansu [18] and Kleiner and Leeb [13] on quasi-isometries of symmetric spaces, and a certain rigidity property of quasi-isometries of hyperbolic spaces proved in Farb and Mosher [5] and Reiter Ahlin [19]. Theorem 2 shows a contrast between the higher-rank case and PSL(2, R), the universal cover of PSL(2, R), for which Asaoka [1] gives (generally) non-trivial orbit-preserving deformations of the actions of AN by right multiplication. Let Ŵ be the flow on Ŵ\ PSL(2, R) defined by the action of A by right multiplication, P the set of oriented periodic orbits of Ŵ , and τ (γ ) the period of γ for γ ∈ P. Consider Ŵ = a ∈ H 1 Ŵ\ PSL(2, R); R sup γ ∈P |a(γ )| τ (γ ) < 1 ,…”

“…We also know how the action ρ a is controlled by the cohomology class a, but we refer the reader to [1] for that and more information. Note that the above deformation is different from the non-orbit-preserving deformation coming from the deformation of the lattice, 4 H. Maruhashi whose deformation space has the dimension equal to that of Teichmüller space, because such deformations are necessarily C 0 volume-preserving.…”