Summary: Infinite Coxeter groups are virtually indicable.
D. Cooper, D. D. Long & A. W. Reid \Lambda
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 3manifolds it remains one of
the outstanding open questions to prove such groups are virtually indicable. To continue on the
3manifold theme, it follows from the work of Hempel  that any closed orientable hyperbolic 3
manifold which admits an orientationreversing involution has fundamental group that is virtually
indicable. In particular if a closed hyperbolic 3manifold M is a finite cover of a hyperbolic 3orbifold
obtained as the quotient of H 3 by a group generated by reflections (i.e. a hyperbolic Coxeter group)
then ß 1 (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 () in connection with Property T (see below):
Theorem 1.1 Let W be an infinite Coxeter group, then W is virtually indicable.
Our methods are motivated from those of lowdimensional topology, in particular the work in
 and  which deal with ``separability properties'' of 3manifold groups.
This theorem has several consequences which seem independently interesting. For example it
Corollary 1.2 Let W be an infinite Coxeter group and K any subgroup of finite index in W . Then