由因子映射和子群胚产生的拓扑全群的最大子群
Maximal subgroups of topological full groups arising from factor maps and subgroupoids
- The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过因子映射和宽开子群胚构造拓扑全群及其换位子群的最大子群，给出最大性判据，并应用于康托尔极小系统，产生 Grigorchuk-Vorobets 猜想中的划分稳定化子及顺从群中的最大子群例子。
AI中文摘要：
我们利用因子映射和具有康托尔单位空间的最小有效 ample 群胚的宽开子群胚，在假设几乎有限性或纯无限性的条件下，构造了拓扑全群及其换位子群的最大子群。对于二对一覆盖和以主目标且纤维基数至多为二的素因子映射，在全群阿贝尔化上的同一诱导映射的满射性和单射性分别刻画了因子全群和因子换位子群的最大性。对于由在单位空间上自由的素数阶循环作用产生的子群胚，较小换位子群在环境全群中的正规化子是一个半直积，当且仅当包含映射在全群阿贝尔化上诱导满射时，该正规化子为最大。在两种构造中，嵌入的换位子群在环境换位子群中具有唯一的最大超群，即其正规化子。对于康托尔极小系统，因子构造产生了 Grigorchuk-Vorobets 猜想中类型的划分稳定化子。子群胚正规化子作用在拓扑上是本原的。群胚同调使得这些系统和有限型移位上的判据可计算。例子包括有限生成的非单且局部有限的单最大子群，它们具有无限指数，位于有限生成的无限单顺从群中。
英文摘要：
We construct maximal subgroups of topological full groups and their commutator subgroups using factor maps and wide open subgroupoids of minimal effective ample groupoids with Cantor unit spaces, assuming almost finiteness or pure infiniteness. For two-to-one coverings and prime factor maps with principal target and fibers of cardinality at most two, surjectivity and injectivity of the same induced map on full-group abelianizations characterize maximality of the factor full group and factor commutator subgroup, respectively. For subgroupoids arising from prime-order cyclic actions free on the unit space, the normalizer of the smaller commutator subgroup in the ambient full group is a semidirect product, maximal exactly when the inclusion induces a surjection on full-group abelianizations. In both constructions, the embedded commutator subgroup has a unique maximal overgroup in the ambient commutator subgroup, namely its normalizer. For Cantor minimal systems, the factor construction yields partition stabilizers of the type in the Grigorchuk-Vorobets conjecture. The subgroupoid normalizers act topologically primitively. Groupoid homology makes the criteria computable for these systems and shifts of finite type. Examples include finitely generated nonsimple and locally finite simple maximal subgroups of infinite index in finitely generated infinite simple amenable groups.
