## Geometry & Topology

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

Hirokazu Maruhashi

#### 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
Revised: 7 August 2015
Accepted: 29 February 2016
First available in Project Euclid: 16 November 2017

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

#### 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

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