## Geometry & Topology

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

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

#### 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.

#### Article information

**Source**

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

**Dates**

Received: 5 May 2014

Revised: 7 August 2015

Accepted: 29 February 2016

First available in Project Euclid: 16 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.gt/1510859131

**Digital Object Identifier**

doi:10.2140/gt.2017.21.157

**Mathematical Reviews number (MathSciNet)**

MR3608711

**Zentralblatt MATH identifier**

1365.37004

**Subjects**

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]

**Keywords**

parameter rigidity solvable Lie groups leafwise cohomology quasiisometry

#### Citation

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. https://projecteuclid.org/euclid.gt/1510859131