Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex