区间同胚的链群
Chain groups of homeomorphisms of the interval
摘要：
本文引入并研究了单流形上同胚的链群的概念，它是Thompson群$F$的某种推广。由此产生的一类群体表现出一致性和多样性的结合。一方面，链群要么有一个简单的换位子群，要么群的作用有一个游荡区间。在后一种情况下，链群允许一个标准商，它也是一个链群，并且它有一个简单的换位子群。另一方面，$\operatorname{Homeo}^+(I)$的每个随机生成的子群可以被实现为链群的子群。作为推论，我们证明了链群有无数个同构类型，$\operatorname{Homeo}^+(I)$的可数单群也有无数个同构类型.我们考虑了不同的链的作用对链群的限制，并证明了有无数的同构类型的$3$-链群不能实现的$C^2$同构，以及无数的同构类型的$6$ -链群不能实现的$C^1$同构.作为推论，我们得到了区间上不存在非平凡$C^1$作用的$\operatorname{Homeo}^+(I)$的单群的无数同构类型.最后，我们证明了如果一个链群极小地作用在区间上，那么它在拓扑共轭下是唯一的。
英文摘要：
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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