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

Abstract

Let $M\stackrel{{\rho }_0}{\curvearrowleft }S$ be a $C^{\infty }$ locally free action of a connected simply connected solvable Lie group $S$ on a closed manifold $M$. Roughly speaking, ${\rho }_0$ is parameter rigid if any $C^{\infty }$ locally free action of $S$ on $M$ having the same orbits as ${\rho }_0$ is $C^{\infty }$ conjugate to ${\rho }_0$. In this paper we prove two types of result on parameter rigidity. First let $G$ be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to ${\mathrm{SO}}_0(n,1)$ $(n\ge 2)$ or $\mathrm{SU}(n,1)$ $(n\ge 2)$, and let $\Gamma$ be an irreducible cocompact lattice in $G$. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu and Kleiner-Leeb on the quasiisometries of Riemannian symmetric spaces of noncompact type. Secondly we show, if $M\stackrel{{\rho }_0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of ${\rho }_0$ with certain coefficients must vanish. This is a partial converse to the results in the author's [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157-191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

39 pages, no figures, published online in Ergodic Theory and Dynamical Systems

References in corpus (2)

• The Large Scale Geometry of Nilpotent-by-Cyclic Groups
• Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups

Cited by in corpus (1)

• Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups