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 denned 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 n copies of the free group π₁ (T), except when T is 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.