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

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