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Morphisms between right-angled Coxeter groups and the embedding problem in dimension two
Mathematische Zeitschrift ( IF 0.8 ) Pub Date : 2021-06-18 , DOI: 10.1007/s00209-021-02794-8
Anthony Genevois

In this article, given two finite simplicial graphs \(\Gamma _1\) and \(\Gamma _2\), we state and prove a complete description of the possible morphisms \(C(\Gamma _1) \rightarrow C(\Gamma _2)\) between the right-angled Coxeter groups \(C(\Gamma _1)\) and \(C(\Gamma _2)\). As an application, assuming that \(\Gamma _2\) is triangle-free, we show that, if \(C(\Gamma _1)\) is isomorphic to a subgroup of \(C(\Gamma _2)\), then the ball of radius \(8|\Gamma _1||\Gamma _2|\) in \(C(\Gamma _2)\) contains the basis of a subgroup isomorphic to \(C(\Gamma _1)\). This provides an algorithm determining whether or not, among two given two-dimensional right-angled Coxeter groups, one is isomorphic to a subgroup of the other.



中文翻译:

直角 Coxeter 群间的态射与二维嵌入问题

在本文中,给定两个有限单纯图\(\Gamma _1\)\(\Gamma _2\),我们陈述并证明了可能态射的完整描述\(C(\Gamma _1) \rightarrow C(\Gamma _2)\)在直角 Coxeter 群\(C(\Gamma _1)\)\(C(\Gamma _2)\) 之间。作为一个应用,假设\(\Gamma _2\)是无三角形的,我们证明,如果\(C(\Gamma _1)\)同构于\(C(\Gamma _2)\)的子群,然后半径的球\(8 | \伽玛_1 || \伽玛_2 | \)\(C(\伽玛_2)\)包含同构子组的基础上\(C(\伽玛_1)\). 这提供了一种算法,用于确定在两个给定的二维直角 Coxeter 群中,一个群是否与另一个的子群同构。

更新日期:2021-06-18
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