The Isomorphism Problem in Coxeter Groups [book]

Patrick Bahls
2005 unpublished
We compute Aut(W ) for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in terms of the given presentation for the Coxeter group and admits an easy characterization of those groups W for which Out(W ) is finite. Reverts to public domain 28 years from publication PATRICK BAHLS then W is called strongly rigid. (In both of these
more » ... both of these cases, any two diagrams V 1 and V 2 for W are isomorphic as edge-labeled graphs.) In case W is infinite and strongly rigid, the group Aut(W ) has a very simple structure (see [6] ): where Diag(W ) consists of the diagram automorphisms of W , those which are induced (in the obvious fashion) by symmetries of the unique diagram V corresponding to W . (When W is finite, this formula may not be true, because Diag(W ) may lie inside Inn(W ), as is the case with the dihedral groups D k for k odd, and the symmetric group Σ k on k letters. Since our concern is primarily with infinite Coxeter groups, we will not concern ourselves with these groups.) The goal of this paper is to describe the automorphism group Aut(W ) of a given even Coxeter group W which satisfies weaker conditions than strong rigidity. This description will admit a finite presentation and will naturally generalize the description given above. Given a Coxeter system (W, S), any element of the form wsw −1 where w ∈ W and s ∈ S is called a reflection of the system (W, S). (This terminology stems from the geometric action of such Coxeter group elements as reflections across hyperplanes in some linear space.) If, for the group W , any two systems (W, S 1 ) and (W, S 2 ) yield the same set of reflections, we call W reflection independent. It is clear that strong rigidity implies reflection independence. More interestingly, if W is even and reflection independent, then it is rigid (see [3] ). We say that a Coxeter system is of large type if the corresponding diagram has no edges labeled 2. Using the main theorem of [3] we conclude that any large-type even Coxeter group is almost always reflection independent, and therefore rigid. These are the groups with which we shall be concerned. (The only obstacle to reflection independence is the presence of "spikes" with label 2(2k +1), k ≥ 1; these are edges [st] in which one of the vertices, say s, has valence 1 in the diagram.) In [2], necessary conditions for an even Coxeter group to be strongly rigid were given, and these conditions were shown to be sufficient provided that W is reflection independent and that either V has no simple circuits of length less than 5 (i.e., no "triangles" and no "squares") or W is of large type. We will mimic the method of proof used in that paper in order to compute Aut(W ) for large-type even groups which additionally satisfy the following condition: (NVB) V contains no vertex s so that V \ {s} consists of more than 2 connected components. If V satisfies this condition, we shall say that it has no vertex branching, or is NVB. We shall also say that W is NVB if its diagram is NVB. Theorem 1.1. Let W be an even, large-type, NVB Coxeter group with connected diagram V. Then Aut(W ) is a semidirect product of G with Diag(W ), where, up to a subgroup of finite index, G is a product of Inn(W ) with certain subgroups of centralizers of edges and vertices of the diagram V.
doi:10.1142/p385 fatcat:5ush2eyy7fdp3nwwla25srvyl4