Words with intervening neighbours in infinite Coxeter groups are reduced

TL;DR

This paper proves that words with the intervening neighbours property in infinite Coxeter groups are irreducible, providing a shorter proof using the root automaton for recognition.

Contribution

It offers a new, shorter proof of irreducibility of certain words in infinite Coxeter groups using the root automaton method.

Findings

Words with the intervening neighbours property are irreducible in infinite Coxeter groups.

A new proof technique using the root automaton is introduced.

The proof simplifies previous arguments for irreducibility.

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