  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.
