自由时间阿贝尔群的Stallings自动机:交集和索引

Pub Date : 2019-10-13 DOI:10.5565/publmat6622209
Jordi Delgado, E. Ventura
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引用次数: 6

摘要

我们将经典的Stallings理论(将自由群的子群描述为自动机)推广到自由群和阿贝尔群的直接乘积:在引入富自动机(即,带有额外阿贝尔标签的自动机)之后,我们得到了子群和某一类型的富自动机之间的显式双射,正如它发生在自由群中一样,在有限生成的情况下是可计算的。这种方法提供了一种简洁的几何描述(甚至是非有限生成的)在这个非howson族中有限生成子群的交集。特别地,我们给出了子群交问题和有限指数问题的几何解,并分别给出了递归基和截线。
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Stallings automata for free-times-abelian groups: intersections and index
We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively.
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