Sarah Rees

A quadratic time solution to the word problem for rank 3 Artin groups

My very recent work with Derek Holt completes a proof that every rank 3 Artin group has word problem that is soluble in quadratic time (the solubility of rank 3 groups was already known; it is method and the complexity bound that is new). This result is a corollary of a collection of results, of Dehornoy&Paris in 1999, myself and Holt in 2012 and 2013, Blasco, Cumplido and Morris-Wright in 2022, as well as my very recent work with Holt, which between them cover all the rank 3 Artin groups.
I’ll start with a brief introduction to Artin groups, a large and diverse family of groups defined by their presentations, which includes the braid groups, free groups and free abelian groups. I’ll define the word problem, and ex- plain what is known about its solution for Artin groups, in particular using rewriting techniques. Then I’ll explain the method of solution which Holt and I found for all sufficiently large Artin groups, how Blasco, Cumplido and Morris-Wright generalised our ideas to cover the large class of 3-free groups, and how Holt and I have now generalised that further to cover particular rank 3 groups, and might hope to get further still. I’ll explain also how recent results apply to an investigation of Deligne complexes.