International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 976390, 10 pages
doi:10.1155/2008/976390
Automorphisms of right-angled Coxeter groups.
Mauricio Gutierrez1
and Anton Kaul2
1Department of Mathematics, Tufts University, Medford, MA 02155, USA
2Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA
Abstract
If (W,S) is a right-angled Coxeter system, then Aut(W) is a semidirect product of the group Aut∘(W) of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut∘(W) is a semidirect product of Inn(W) by the quotient Out∘(W)=Aut∘(W)/Inn(W). We also give sufficient conditions for the compatibility of the two semidirect products. When this occurs there is an induced splitting of the sequence 1→Inn(W)→Aut(W)→Out(W)→1 and consequently, all group extensions 1→W→G→Q→1 are trivial.