On the metabelian property of quotient groups of solvable groups of orientation-preserving homeomorphisms of the line

TL;DR

This paper proves that for certain solvable groups of orientation-preserving homeomorphisms of the line, the quotient group formed by dividing out stabilizers of the minimal set is metabelian, aiding in classifying such groups.

Contribution

It establishes the metabelian property of the quotient group for a class of solvable groups of homeomorphisms, advancing the understanding of their structure and classification.

Findings

The quotient group G/H_G is metabelian.

The result applies to solvable groups with a freely acting element.

This contributes to the classification of groups related to Thompson's group F.

Abstract

For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are stabilizers of the minimal set. This fact is an important element in the classification theorem, used, in particular, in the study of the Thompson's group $F$.