On a family of finitely generated simple groups of homeomorphisms of the circle

Abstract

The notion of chain groups of homeomorphisms of $\mathbb{R}$ was introduced by Kim, Koberda and Lodha as a generalization of Thompson's group $F$. Subsequently, an $S^1$-version of chain groups, known as ring groups, has been studied. In this paper, we further 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 $\mathbb{R}$-trees. We give a construction of ring groups such that for every fixed point-free isometric action on an $\mathbb{R}$-tree, there exists an invariant line upon which the group acts by translations. In other words, such ring groups have property A$\mathbb{R}$. We also confirm that there exist uncountably many finitely generated simple groups in the group of orientation preserving homeomorphisms of $S^1$, which are commutator subgroups of ring groups.