We show that if is a solvable group acting on the line and if there is having no fixed points, then there is a Radon measure on the line quasi-invariant under . In fact, our method allows for the same conclusion for inside a class of groups that is closed under extensions and contains all solvable groups and all groups of subexponential growth.
"Quasi-invariant measures for some amenable groups acting on the line." Algebr. Geom. Topol. 18 (2) 1067 - 1076, 2018. https://doi.org/10.2140/agt.2018.18.1067