Coxeter covers of the symmetric groups

Details Coxeter covers of the symmetric groups Coxeter covers of the symmetric groups

2004-05-11

arxiv.org/abs/math/0405185v1

Mathematics

Group Theory

32 pp. Accepted to Journal of Group Theory

Scientific paper

We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set $T$ of transpositions. These quotients, denoted here by C_Y(T), are a special type of the generalized Coxeter groups defined in \cite{CST}, and also arise in the computation of certain invariants of surfaces. We use a surprising action of $S_n$ on the kernel of the surjection $C_Y(T) \ra S_n$ to show that this kernel embeds in the direct product of $n$ copies of the free group $\pi_1(T)$ (with the exception of $T$ being the full set of transpositions in $S_4$). As a result, we show that the groups $C_Y(T)$ are either virtually Abelian or contain a non-Abelian free subgroup.

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