Study Geometric Group Theory at McGill

This subject will be attractive to students who are intrigued by topology, algebra, or geometry.

We have many exciting projects on geometric group theory. Below are several possibilities, each of which can be pursued at either the MA or Ph.D. level.

Request an application by E-mail and state that you are interested in studying geometric group theory.

The geometry of nonpositively curved cubulated spaces.
Let X be a space which is formed by gluing together a bunch of cubes along their faces. If we rule out certain "bad gluings" then the space X will be "nonpositively curved". There are a great many problems related to these cubulated spaces. Some of these have potential important applications towards 3-manifold topology.

The fundamental groups of knot complements.
Let K be a knotted circle in R3, and let M = R3 - be its complement. The goal of this project is to construct certain "splittings" of M which determine analagous splittings of the fundamental group of M. These splittings will

The residual finiteness of one-relator groups with torsion.
Gilbert Baumslag conjectured in 1968 that every one-relator group with torsion is residually finite.
These are groups presented by < a, b, ... | Wn > where n >1.
This project presents many opportunities for computer study of this problem, and the development of a strategy to prove Baumslag's conjecture in general.

Algebra and geometry of nonpositively curved spaces with isolated spots. (Taken)
This is a project to generalize properties of Gromov's hyperbolic spaces to nonpositively curved spaces with isolated flats. While many negative curvature properties fail to hold in the nonpositively curved context, they will hold if we restrict the flatness to lie in isolated flat planes.

The coherence of one-relator groups.
A group is finitely presented if it can be described using a finite set of generators and relations between those generators and their inverses <a1, a2, ..., am | R1 = 1,  R2 = 1, ..., Rn = 1>.
A group is coherent if each of its finitely generated subgroups is finitely presented.
Remarkably, it is unknown whether every group given by a presentation with a single defining relation is coherent. This project will develop a strategy towards solving this problem!

Algebraic geometry over a free group.
One can define a Zariski topology on  F n taking solution sets of  equations as closed sets.
We plan to develop a notion of dimension in F n.

Equationally Noetherian groups.
A group G is called Equationally Noetherian if every system of equations over G is equivalent to a finite subsystem.
Is a torsion free hyperbolic group Equationally Noetherian ?
Is the universal theory of a torsion free hyperbolic group decidable ?

Fully residually free groups.
A group G is fully residually free is for each finite set of nontrivial
elements there exists a homomorphism from G into a free group such that
the images of these elements are nontrivial.
Finitely generated fully residually free groups are described
by Kharlampovich and Miasnikov (J.Algebra, 98). They have a
very nice structure, act freely on Z n trees. This project involves
investigation of different algorithmic problems for such groups.

Groups acting on non-archimedian trees.
We plan to study and possibly describe groups acting freely on non-archimedian trees.

Amalgamated products of free groups.
We plan to solve different algorithmic problems for such groups.