 
Summary: NORMALIZERS OF PARABOLIC SUBGROUPS OF
COXETER GROUPS
DANIEL ALLCOCK
Abstract. We improve a bound of Borcherds on the virtual co
homological dimension of the nonreflection part of the normalizer
of a parabolic subgroup of a Coxeter group. Our bound is in terms
of the types of the components of the corresponding Coxeter sub
diagram rather than the number of nodes. A consequence is an
extension of Brink's result that the nonreflection part of a re
flection centralizer is free. Namely, the nonreflection part of the
normalizer of parabolic subgroup of type D5 or Am odd is either
free or has a free subgroup of index 2.
Suppose is a Coxeter diagram, J is a subdiagram and WJ
W is the corresponding inclusion of Coxeter groups. The normalizer
NW
(WJ ) has been described in detail by Borcherds [3] and Brink
Howlett [5]. Such normalizers have significant applications to working
out the automorphism groups of Lorentzian lattices and K3 surfaces;
see [3] and its references. NW
(WJ ) falls into 3 pieces: WJ itself, another
