Artin groups are a large class of groups commonly studied by geometric group theorists. They are related to Coxeter groups, they include and generalize the braid groups, and they have a natural definition involving complexified hyperplane complements. Despite these affiliations with well-known objects, very few Artin groups are currently understood, even at a basic level, and those that are belong to highly restrictive families defined primarily by the situations where existing techniques can be applied. The approach described here is fundamentally new and different. The goal of this proposal is to understand the natural, continuous geometric objects (recently introduced by the principal investigator and his collaborators) that have the potential to serve as a geometric foundation for the study of all Artin groups in a uniform fashion. These geometric objects, called factor geometries, are of interest in their own right. There is, for example, a natural continuous group G that has the orthogonal group as a quotient, contains the braid group as a subgroup, and is the fundamental group of a finite-dimensional metric simplicial complex whose universal cover is the contractible factor geometry with a building-like structure on which G acts. This unusual uncountable group and space, defined via the collection of minimal factorizations of spherical isometries into reflections, is a hybrid mix of Lie groups and metric simplicial complexes, and every Artin group acts on a similarly defined continuous geometric object. The eventual structure theory should resemble that of Lie groups and early indications are that these types of complexes carry geometric and combinatorial structures sufficient to resolve many longstanding conjectures about arbitrary Artin groups.
Mathematical objects, like many physical objects, can be better understood when we fully understand the symmetries they possess. The algebraic structure that records how these symmetries interact is called a ``group'' and the groups under consideration here are a class of groups generated by ``reflections'' (a symmetry like the reflected image one sees through a mirror). The main goal of this project is to deepen our understanding of the relationship between symmetry groups built from reflections (specifically Coxeter groups) and a second class of symmetry groups, called Artin groups. The most famous example of an Artin group is the braid group, the group that keeps track of the distinct ways in which several strands of string can be braided together. Constructions involving Coxeter groups, Artin groups and braid groups proliferate throughout mathematics, including some recent mathematical physics. In many of these cases, the immediate connections are, strictly speaking, to Artin groups rather than Coxeter groups, but these connections have not been pursued partly because the theory of Artin groups is underdeveloped. Once this situation is rectified within geometric group theory, the next step will be to export the resulting structure theory to these neighboring domains
