Chain groups of homeomorphisms of the interval and the circle

We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson’s group F. Precisely, this is a group finitely generated by homeomorphisms, each supported on exactly one interval in a chain, and subject to a certain mild dynamical condition. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. Moreover, any 2-chain group is isomorphic to Thompson’s group F. On the other hand, every finitely generated subgroup of Homeo pIq can be realized as a subgroup of a chain group. As a corollary, we show that there are uncountably many isomorphism types of chain group, as well as uncountably many isomorphism types of countable simple subgroups of Homeo pIq. We also study the restrictions on chain groups imposed by actions of various regularities, and show that there are uncountably many isomorphism types of chain groups which cannot be realized by C2 diffeomorphisms.