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 $\mathrm{SO}_0(n,1)$ $(n\geq2)$ or $\mathrm{SU}(n,1)$ $(n\geq2)$, 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.