Infinite Coxeter Groups Are Virtually Indicable

An infinite group G is called indicable (resp. virtually indicable) if G (resp. a subgroup of finite index in G) admits a homomorphism onto Z. This is a powerful property for a group to have; for example in the context of infinite fundamental groups of aspherical 3-manifolds it remains one of the outstanding open questions to prove such groups are virtually indicable. To continue on the 3-manifold theme, it follows from the work of Hempel [8] that any closed orientable hyperbolic 3-manifold which admits an orientation-reversing involution has fundamental group that is virtually indicable. In particular if a closed hyperbolic 3-manifold M is a finite cover of a hyperbolic 3orbifold obtained as the quotient of H by a group generated by reflections (i.e., a hyperbolic Coxeter group) then nt(M) is virtually indicable. The purpose of this note is to prove the following theorem, posed as a question by P. De La Harpe and A. Valette ([5]) in connection with Property T (see below):