Clara Flegel

Project Title: The non-positively curved geometry of Artin groups

Algebra involves the study of mathematical symbols and the rules for manipulating these symbols. Geometry, on the other hand, is concerned with properties of space such as shapes, size and distance. The area of geometric group theory combines both of these branches of mathematics by considering mathematical objects called groups, which are defined by their algebraic rules, and treating them as geometric objects. The focus of this project will be a type of groups called Artin groups, which have a similar algebraic definition to the rules one would describe for obtaining braided strands of hair.

Similar to how one can turn a cube in their hand to study its symmetries from different sides, we can study groups by realising them as the symmetries of a nice geometric object. One can often learn a lot about the algebraic and geometric structure by studying the symmetries of the group on complexes naturally associated to them. A particularly interesting complex associated to an Artin group is called its Deligne complex. This object is obtained by considering special building blocks of the group and how they are contained within one another.

The overall goal of this project will be to explore the structure and geometry of certain Artin groups, which is still mysterious in general. One of the main objectives along this road will be to show that the Deligne complex of certain three-dimensional Artin groups is indeed a metric space of non-positive curvature, or even negative curvature, in certain cases.