Chain groups of homeomorphisms of the interval

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. 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. 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 groups, as well as uncountably many isomorphism types of countable simple subgroups of Homeo`pIq. We consider the restrictions on chain groups imposed by actions of various regularities, and show that there are uncountably many isomorphism types of 3–chain groups which cannot be realized byC2 diffeomorphisms, as well as uncountably many isomorphism types of 6–chain groups which cannot be realized byC1 diffeomorphisms. As a corollary, we obtain uncountably many isomorphism types of simple subgroups of Homeo`pIq which admit no nontrivialC1 actions on the interval. Finally, we show that if a chain group acts minimally on the interval, then it does so uniquely up to topological conjugacy.