On subgroups with narrow Schreier graphs

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-10-04 DOI:10.1112/blms.13157

Pénélope Azuelos

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Abstract

We study finitely generated pairs of groups $H ⩽ G$ such that the Schreier graph of $H$ has at least two ends and is narrow. Examples of narrow Schreier graphs include those that are quasi-isometric to finitely ended trees or have linear growth. Under this hypothesis, we show that $H$ is a virtual fiber subgroup if and only if $G$ contains infinitely many double cosets of $H$ . Along the way, we prove that if a group acts essentially on a finite-dimensional CAT(0) cube complex with no facing triples, then it virtually surjects onto the integers with kernel commensurable to a hyperplane stabiliser.

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关于窄Schreier图的子群

我们研究了有限生成的群对H≤G $H \leqslant G$，使得H $H$的Schreier图至少有两个端点并且是窄的。窄Schreier图的例子包括那些有限端树的准等距图或线性增长图。在此假设下，我们证明H $H$是虚光纤子群当且仅当G $G$包含无穷多个H $H$的双余集。在此过程中，我们证明了如果一个群本质上作用于一个没有面向三元组的有限维CAT(0)立方复合体上，那么它实际上投射到核可与超平面稳定子通约的整数上。

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