The space of relative orders and a generalization of Morris indicability theorem

TL;DR

This paper introduces the concept of relative orders on groups, proves their compactness for finitely generated groups, and generalizes Morris's indicability theorem to broader group actions on the line.

Contribution

It defines the space of relative orders and demonstrates its compactness, extending Morris's theorem to include groups with certain stabilizer properties.

Findings

The space of relative orders is compact for finitely generated groups.

Groups acting on the line with proper co-amenable stabilizers surject onto z.

Generalization of Morris's indicability theorem to new group action contexts.

Abstract

We introduce the space of relative orders on a group and show that it is compact whenever the group is finitely generated. We use this to show that if $G$ is a finitely generated group acting by order preserving homeomorphism of on the line, then if some stabilizer of a point is proper and co-amenable subgroup, then $G$ surjects onto $\mathbb{Z}$. This is a generalization of a theorem of Morris.