Automorphisms of Coxeter Groups of Rank 3 with Infinite Bonds

Abstract

If W is a rank 3 Coxeter group, whose Coxeter diagram contains at least one infinite bond, then the automorphism group of W is larger than the group generated by the inner automorphisms and the automorphisms induced from automorphisms of the Coxeter diagram. In each case the automorphism group of W is described. PRELIMINARIESThe aim of this paper is to describe the automorphism group of rank 3 Coxeter groups whose diagram contains at least one infinite bond.We recall that a Coxeter group is a group with a presentation of the form W = gp r a a ∈ r a r b m ab = 1 for all a b ∈ where is some indexing set, whose cardinality is called the rank of W , and the parameters m ab satisfy the following conditions: m ab = m ba , each m ab lies in the set m ∈ m ≥ 1 ∪ ∞ , and m ab = 1 if and only if a = b. To simplify matters write r i for the generator r a i . If w ∈ W then we define l w to be the length of the shortest expression for w as a product of generators r a (a ∈ ).The (Coxeter) diagram of W is a graph with vertex set in which an edge (or bond) labeled m ab joins a b ∈ whenever m ab ≥ 3. We say that the group is irreducible if this graph is connected.Let V be a real vector space with basis , and define a bilinear form B on V by B a b = − cos π/m ab for all a b ∈ with m ab < ∞ and B a b = −1 otherwise. For each a ∈ V such that B a a = 1 we define

Involution words II: braid relations and atomic structures

“…In this situation we write w| J and w| K for the images of w ∈ W under the natural projections W → W J and W → W K , so that w = w| J · w| K = w| K · w| J . We will need the following fact: Proposition 3.1 (Franzsen and Howlett [8]). If v ∈ W then v −1 sv ∈ S for all s ∈ S if and only if v = w J for a subset J ⊂ S which commutes with S \ J and which is such that W J is finite.…”

“…This implies that (a) and (b) hold by Proposition 2. 8. Alternatively suppose us < u; one of the following cases then occurs:…”

Automorphisms of nearly finite Coxeter groups

“…The automorphism groups of infinite rank 3 Coxeter groups whose diagrams have no edges with infinite labels are described in [9]; in this case the automorphism group is the semi-direct product of Inn(W ) and the group of graph automorphisms. The automorphism groups of rank 3 Coxeter groups with both finite and infinite edge labels are described in [7].For the purposes of this paper, we say that an infinite Coxeter group is nearly finite if it has finite rank n and has a finite parabolic subgroup of rank n − 1. It is shown that if W is nearly finite and does not have an edge labelled ∞ then the group of all automorphisms of W that preserve reflections is the semi-direct product of Inn(W ) and the group of graph automorphisms.…”

“…Finally, in [14], Mühlherr gave a presentation for the automorphism group of any right-angled Coxeter group. The automorphism groups of infinite rank 3 Coxeter groups whose diagrams have no edges with infinite labels are described in [9]; in this case the automorphism group is the semi-direct product of Inn(W ) and the group of graph automorphisms. The automorphism groups of rank 3 Coxeter groups with both finite and infinite edge labels are described in [7].…”

“…The automorphism groups of rank 3 Coxeter groups with both finite and infinite edge labels are described in [7].…”