Louis Rowen, Mina Teicher, Uzi Vishne
July 27, 2005
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 , 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 π 1 ( T ), except when T is the full set of transpositions in S 4 . As a result, we show that each group C Y ( T ) either is virtually Abelian or contains a non-Abelian free subgroup.