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

TL;DR

This paper proves the existence of quasi-invariant Radon measures for certain solvable and amenable groups acting on the real line, extending to a broader class of groups with specific properties.

Contribution

It establishes the existence of quasi-invariant measures for a class of groups acting on the line, including solvable groups and those with subexponential growth.

Findings

Existence of Radon measures quasi-invariant under group actions.

Extension of results to groups closed under extensions with specific properties.

Applicable to groups with no fixed points for some element.

Abstract

In this note we show that if $G$ is a solvable group acting on the line, and if there is $T\in G$ 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.