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Louis Rowen, Mina Teicher, Uzi Vishne

Coxeter covers of the symmetric groups

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 case of the generalized Coxeter groups defined in [5], 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 ) ? S_n to show that this kernel embeds in the direct product of ncopies of the free group ? ₁(T ), except when Tis the full set of transpositions in S₄. As a result, we show that each group C_Y(T ) either is virtually Abelian or contains a non-Abelian free subgroup.

Journal of Group Theory, Walter de Gruyter

Print ISSN: 1433-5883
Volume: 8, 03/2005
Pages: 139 - 169

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