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

TL;DR

This paper investigates parameter rigidity of certain solvable Lie group actions on manifolds, proving rigidity results for specific actions of semisimple Lie groups and exploring cohomological conditions linked to rigidity.

Contribution

It establishes parameter rigidity for actions of certain lattices in semisimple Lie groups and links cohomology vanishing to parameter rigidity of solvable Lie group actions.

Findings

Proves parameter rigidity of $ ext{AN}$ actions in semisimple Lie groups with specific conditions.

Shows vanishing of zeroth and first cohomology is necessary for parameter rigidity.

Connects rigidity results with quasiisometry theorems of symmetric spaces.

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.