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(+) (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 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 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.
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