The notion of chain groups of homeomorphisms of 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 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. In other words, such ring groups have property A 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.
Motoko Kato (Wed,) studied this question.