Structure theorems for subgroups of homeomorphisms groups

TL;DR

This paper explores the structure of certain groups of circle homeomorphisms that lack non-abelian free subgroups, providing new classifications and proofs within the context of dynamical systems and group theory.

Contribution

It offers a detailed classification of solvable subgroups of R. Thompson's group T and presents new proofs and results about groups of circle homeomorphisms.

Findings

Classification of solvable subgroups of R. Thompson's group T

New proof of Margulis's theorem

Extension of classical results on circle homeomorphisms

Abstract

In this partly expository paper, we study the set A of groups of orientation-preserving homeomorphisms of the circle S^1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in A. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T .