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

TL;DR

This paper establishes conditions under which actions of solvable Lie groups are parameter rigid, using vanishing cohomology and quasiisometry properties, with applications to actions on mapping tori and relations to lattice theory.

Contribution

It introduces a new sufficient condition for parameter rigidity of solvable Lie group actions based on cohomology vanishing and explores applications to specific group actions and lattice relations.

Findings

Vanishing of uncountably many first cohomologies implies parameter rigidity.

Finiteness of cohomology vanishing can suffice in some cases.

Applications include proving rigidity of actions on mapping tori.

Abstract

We give a sufficient condition for parameter rigidity of actions of solvable Lie groups, by vanishing of (uncountably many) first cohomologies of the orbit foliations. In some cases, we can prove that vanishing of finitely many cohomologies is sufficient. For this purpose we use a rigidity property of quasiisometry. As an application we prove some actions of 2-step solvable Lie groups on mapping tori are parameter rigid. Special cases of these actions are considered in a paper of Matsumoto and Mitsumatsu. We also remark on the relation between transitive locally free actions of solvable Lie groups and lattices in solvable Lie groups, and apply results in rigidity theory of lattices in solvable Lie groups to construct transitive locally free actions with some properties.