Words with Intervening Neighbours in Infinite Coxeter Groups are Reduced

Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of irreducible words. 1. Words with intervening neighbours Let G be the Coxeter graph of a Coxeter group with generators S. Consider a word w in the alphabet S. Definition 1.1. A word has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. In the example below, s0s1s0s2 has this property, but s0s1s0s2s1 lacks it, since the two occurrences of s1 are not separated by the neighbour s3. m s0 m s1 m s3 m s2 @ David Speyer [7] recently proved the following result. Theorem 1.2. For an infinite irreducible Coxeter group, all words with the intervening neighbours property are irreducible. In this note, we will demonstrate how the proof of this general result can be reduced to checking the property for just the affine Coxeter groups and just a small subset of words, for which verification of the property is straightforward. Our tool will be the finite automaton for recognition of irreducible words, invented in [4]. 2. The root automaton For any group given by generators and relations, a word w in the generators is called irreducible if it is the shortest word for that group element. In general, recognizing irreducible words is an undecidable problem. For a Coxeter group, however, a finite recognizing automaton exists [3]. We will here use the concrete root automaton developed by H. Eriksson (for details, see [1]). In brief, a root in a Coxeter group can be represented as a vector of numbers, one for each vertex of the Coxeter graph. Let mxy ≥ 3 denote the label of the edge between two neighbouring vertices x and y in the Coxeter