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.
More details about graduate study at McGill can be found
here.
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 R^{3}, and let M
= R^{3} - K 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, ... | W^{n}
> 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 <a_{1}, a_{2}, ..., a_{m}
| R_{1} = 1, R_{2 }= 1, ...,
R_{n}
= 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.