Coxeter covers of the symmetric groups

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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 CY(T), are a special type of the generalized Coxeter groups defined in CST, and also arise in the computation of certain invariants of surfaces. We use a surprising action of Sₙ on the kernel of the surjection CY(T) Sₙ to show that this kernel embeds in the direct product of n copies of the free group π₁(T) (with the exception of T being the full set of transpositions in S₄). As a result, we show that the groups CY(T) are either virtually Abelian or contain a non-Abelian free subgroup.