Volume 21, issue 1 (2017)

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

Hirokazu Maruhashi

Geometry & Topology 21 (2017) 157–191

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.

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parameter rigidity, solvable Lie groups, leafwise cohomology, quasiisometry
Mathematical Subject Classification 2010
Primary: 37A20
Secondary: 37C15, 37C85
Received: 5 May 2014
Revised: 7 August 2015
Accepted: 29 February 2016
Published: 10 February 2017
Proposed: Martin Bridson
Seconded: Walter Neumann, Benson Farb
Hirokazu Maruhashi
Max Planck Institute for Mathematics
Vivatsgasse 7
D-53111 Bonn