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 $\operatorname{Homeo}^+(I)$ 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 $\operatorname{Homeo}^+(I)$. 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 by $C^2$ diffeomorphisms, as well as uncountably many isomorphism types of $6$--chain groups which cannot be realized by $C^1$ diffeomorphisms. As a corollary, we obtain uncountably many isomorphism types of simple subgroups of $\operatorname{Homeo}^+(I)$ which admit no nontrivial $C^1$ 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.