Quasi-invariant measures for some amenable groups acting on the line

We show that if G is a solvable group acting on the line and if there is T is an element of G having no fixed points, then there is a Radon measure mu on the line quasi-invariant under G. In fact, our method allows for the same conclusion for G inside a class of groups that is closed under extensions and contains all solvable groups and all groups of subexponential growth.