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Reflections in abstract Coxeter groups

W. N. Franzsen·R. B. Howlett·B. Mühlherr
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Abstract

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite sugroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.

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W. N. Franzsen, R. B. Howlett, B. Mühlherr. 2005-06-28. Reflections in abstract Coxeter groups. https://arxiv.org/abs/math/0506573

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