Reflections in abstract Coxeter groups
W. N. Franzsen, R. B. Howlett, B. Mühlherr
Abstract
Let W be a Coxeter group and r∈ 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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