Rewriting in Artin groups without A_3 or B_3 subdiagrams

TL;DR

This paper demonstrates that the word problem in certain Artin groups, specifically those without A_3 or B_3 subdiagrams, can be efficiently solved using length-preserving rewrite rules, reducing words to geodesics in quadratic time.

Contribution

It extends existing results by providing a quadratic-time solution for the word problem in a new class of Artin groups defined by their diagram restrictions.

Findings

Quadratic time reduction to geodesic words

Length-preserving rewrite rules are effective

Applicable to Artin groups without A_3 or B_3 subdiagrams

Abstract

We prove that the word problem in an Artin group G based on a diagram without A_3 or B_3 subdiagrams can be solved using a system of length preserving rewrite rules which, together with free reduction, can be used to reduce any word over the standard generators of G to a geodesic word in G in quadratic time. This result builds on work of Holt and Rees, and of Blasco-Garc\'ia, Cumplido and Morris-Wright. Those articles prove the same result for all Artin groups that are either sufficiently large or 3-free, respectively.