UC Santa Barbara

In 1925 Emil Artin introduced the braid groups, a widely studied class of groups which have applications in many mathematical and non-mathematical fields. These were later generalized to a larger class of groups, now called Artin groups, which have elegant, finite presentations.Despite these nice presentations, the word problem for most Artin groups has not been solved. The word problem, which seeks to algorithmically determine when two words represent the same group element, has been studied for over a century and a positive solution is essential to any systematic computational study of an infinite discrete group.

In 1972 Brieskorn and Saito showed that the spherical Artin groups have a solvable word problem using, in modern terminology, Garside structures. In 2017 Jon McCammond and Robert Sulway showed that Euclidean Artin groups also have a decidable word problem using the dual presentations of Euclidean Artin groups. These dual Euclidean Artin groups are isomorphic to their corresponding Euclidean Artin groups, but their group presentations include an infinite generating set. This infinite generating set gives some dual Euclidean Artin groups a Garside structure which provides a nice solution to the word problem.

In this dissertation we study the algorithm that solves the word problem in the dual Euclidean Artin group of type $\textsc{Art}(\widetilde{A}_2)$ using its infinite generating set. We then rework this algorithm to solve the word problem for $\textsc{Art}(\widetilde{A}_2)$ using its standard, finite generating set.