Coxeter covers of the symmetric groups

2004/05/11 by Louis H. Rowen, Rowen, Louis H., Mina Teicher +3
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #math.AG #math.GR

paper · pdf · doi:10.48550/arxiv.math/0405185

32 pp. Accepted to Journal of Group Theory

arxiv created 2004/05/11 · arxiv updated 2009/12/01

Abstract

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 \citeCST, 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 π₁(T) (with the exception of T being the full set of transpositions in S₄). As a result, we show that the groups C_Y(T) are either virtually Abelian or contain a non-Abelian free subgroup.

Related

• Coxeter covers of the classical Coxeter groups
• Symmetric Presentations of Coxeter Groups
• Coxeter groups preserving orthants
• Simply-laced Coxeter groups and groups generated by symplectic transvections
• On non-conjugate Coxeter elements in well-generated reflection groups