Geometry & Topology

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

Hirokazu Maruhashi

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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.

Article information

Geom. Topol., Volume 21, Number 1 (2017), 157-191.

Received: 5 May 2014
Revised: 7 August 2015
Accepted: 29 February 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations
Secondary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]

parameter rigidity solvable Lie groups leafwise cohomology quasiisometry


Maruhashi, Hirokazu. Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21 (2017), no. 1, 157--191. doi:10.2140/gt.2017.21.157.

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