Solvable Groups Acting on the Line

Abstract

Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also determined for the existence of invariant (quasi-invariant) measures for abelian (solvable) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of ${C^2}$ diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated.