Pure Salvetti complexes and Euclidean Artin groups
Coxeter groups are a general family of groups that contain the isometry groups of the Platonic solids and the symmetry groups of regular Euclidean tilings. These groups are ubiquitous and well-understood. They are also closely linked to the lesser-known braided versions known as Artin groups. In this dissertation, I investigate the word problem for Artin groups corresponding to Coxeter groups that act naturally on Euclidean space. The corresponding Artin group is the fundamental group of the quotient of the complexified Euclidean space after removing the fixed hyperplanes of the reflections in the Coxeter group.
To understand a Euclidean Artin group, I focus on the structure of the infinite sheeted cover corresponding to the kernel of the homomorphism from the Artin group to the Coxeter group. This space deformation retracts onto a complex constructed as an oriented version of the complex dual to the tessellation preserved by the Coxeter group. This is a multivertex complex with infinitely many vertices. The fundamental groups of its subcomplexes have not been previously studied. The subcomplexes where the inclusion map induces an injection on fundamental groups are of particular interest. This condition is known as $\pi_1$-injectivity.
Given a sufficiently rich family of compact $\pi_1$-injective subcomplexes, the word problem for the full Artin group can be reduced to the word problem for the fundamental groups of the subcomplexes in this family. The goal is to produce such families of subcomplexes, and then to reduce the word problem of their fundamental groups to the word problem of a finite list of "atomic" subcomplexes.
In this dissertation I present a solution to the word problem, using this approach, for the Artin groups corresponding to the infinite dihedral group, the 333 triangle group, and 244 triangle group. And I describe the difficulties that one encounters when trying to extend these methods to the Artin group corresponding to the 236 triangle group or to other higher dimensional Euclidean Artin groups.