Artin groups of type (2,3,n)

Abstract

Our main theorem is that the word problem in the Artin group G = <a,b,c | aba=bab, ac=ca, {}_{n}(b,c) = {}_{n}(c,b) > for n >= 5 can be solved using a system R 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. Since every rank 3 Artin group is either spherical or in one of the categories covered by the previous results on which we build, it follows that any rank 3 Artin group has quadratic Dehn function. However we note that this and much more is a consequence of very recent work of Haettel and Huang; our contribution is to provide a particular kind or rewriting solution to the word problem for the non-spherical rank 3 Artin groups (and more).