Hierarchies and graph products of elementary groups

This project is at the intersection of the mathematical fields of algebra and geometry. We study algebraic objects called groups. These are the algebraic abstractions of symmetry groups of geometric objects. In order to consider an abstract group as a symmetry group, we need to construct a suitable geometric object on which the group can act. The geometry of this object can provide information about the algebraic structure of the group. The particular group of interest in this project are those that can be built from the simplest groups through two different possible group combining operations. In this way we can form complex groups, although they are made up of the simplest possible parts. One might expect that the corresponding geometric objects could be constructed in a similar way from simple pieces. This is true, but it turns out that sometimes the same geometries can be created in different ways. One of our main goals is to show that such a geometric coincidence is due to the fact that the two groups in question are actually closely related.

This project is about interactions between Algebra and Geometry. We studied Algebraic objects called Groups, which are the symmetries of some Geometric Space. We were particularly interested in a construction called Graph Products of Elementary Groups, which describes a Group built from the simplest of pieces according to local information encoded by a Graph. This class includes two very well known families called right-angled Artin Groups and right-angled Coxeter groups. Much of our work focused on determining when groups from these two families have similar Geometries.