A new construction of uncountably many finitely generated simple groups of homeomorphisms of the circle

TL;DR

This paper constructs uncountably many finitely generated simple groups of circle homeomorphisms using ring groups, extending chain groups, and analyzes their simplicity and actions on R-trees.

Contribution

It introduces ring groups as an $S^1$-version of chain groups, proves their commutator subgroups are simple under certain conditions, and constructs uncountably many such simple groups.

Findings

Uncountably many finitely generated simple groups of circle homeomorphisms are constructed.

The simplicity of commutator subgroups of ring groups is established under minimal action conditions.

Ring groups can have invariant lines in their actions on R-trees, with some acting by translations.

Abstract

The notion of chain groups of homeomorphisms of the interval was introduced by Kim, Koberda and Lodha as a generalization of Thompson's group $F$. In this paper, we study an $S^1$-version of chain groups: ring groups. We study the simplicity of the commutator subgroups of ring groups. We show that a ring group with a prechain subgroup acting minimally on its support has a simple commutator subgroup. We also study isometric actions of ring groups on R-trees. We give a construction of ring groups such that for every fixed point-free isometric action on an R-tree, there exists an invariant line upon which the group acts by translations. We also confirm that there are uncountably many finitely generated simple groups in the group of orientation preserving homeomorphisms of $S^1$, which are commutator groups of ring groups.