Words with Intervening Neighbours in Infinite Coxeter Groups are Reduced

Abstract

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 reduced if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of reduced words.