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Infinite Coxeter groups are virtually indicable. D. Cooper, D. D. Long & A. W. Reid \Lambda

Summary: Infinite Coxeter groups are virtually indicable.
D. Cooper, D. D. Long & A. W. Reid \Lambda
1 Introduction.
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 [8] 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 ([5]) 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
[9] and [10] 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


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics