Artin groups of type (2,3,n)

Derek F. Holt, Sarah Rees

Published 2023-10-23Version 1

We complete the proof that every rank 3 Artin group has quadratic word problem. This result follows from the main theorem of this article, together with results of Deligne, of Holt and Rees, and of Blasco, Cumplido and Morris-Wright. Our main theorem is that the word problem in the Artin group G= < a,b,c \mid aba=bab, ac=ca, {}_{n}(b,c) = {}_{n}(c,b) > for n at least 5 can be solved using a system of length preserving rewrite rules that, together with free reduction, can be used to reduce any word over {a,b,c} to a geodesic word in $G$, in quadratic time. This result builds on work of Holt and Rees, and of Blasco, Cumplido and Morris-Wright, which proves the same result for all Artin groups that are either sufficiently large or 3-free.