Words with intervening neighbours in infinite Coxeter groups are reduced. (Q2380420)

Summary: 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. \textit{D. E. Speyer} recently proved that words with the intervening neighbours property are reduced if the group is infinite and irreducible [Proc. Am. Math. Soc. 137, No. 4, 1295-1302 (2009; Zbl 1187.20053)]. We present a new and shorter proof using the root automaton for recognition of reduced words.