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Morphisms between right-angled Coxeter groups and the embedding problem in dimension two

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Abstract

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.

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Acknowledgements

I am grateful to Ivan Levcovitz and Sang-Hyun Kim for their comments on an earlier version of the article. This work was supported by a public grant as part of the Fondation Mathématique Jacques Hadamard.

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  1. Universite de Montpellier, Montpellier, France

    Anthony Genevois

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  1. Anthony Genevois
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Correspondence to Anthony Genevois.

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Genevois, A. Morphisms between right-angled Coxeter groups and the embedding problem in dimension two. Math. Z. 300, 259–299 (2022). https://doi.org/10.1007/s00209-021-02794-8

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