Sean O'Brien - University of Glasgow

The dual Artin group is a group construction coming from an interval of reflection factorisations in Coxeter groups, and is conjectured to be isomorphic to the corresponding (standard) Artin group. It has played a key role in the last decade in advancing some problems related to Artin groups, especially those of affine type, where it was vital in providing a solution to the word problem and proving the K(\pi,1) conjecture.
We explore the construction and the isomorphism conjectured above. We give necessary and sufficient conditions for the isomorphism, and argue why these hold when all defining relations in the Coxeter group are sufficiently long e.g. abab = baba (length 4), or longer. Our methods use simple arguments about embedded paths in the plane, and exploit how these interact and constrain the solution to the word problem in Coxeter groups.