Coarse homotopy groups and large scale geometry of groups with bi-invariant metrics

These projects are open to students worldwide, but have no funding attached. Therefore, the successful applicant will be expected to fund tuition fees at the relevant level (home or international) and any applicable additional research costs. Please consider this before applying.

Following Gromov, this project is concerned with coarse geometry of metric spaces. The slogan one should have in mind is: “things at bounded distance are equivalent”, in which case we call them “coarsely equivalent” or “large scale equivalent”. For example,

• Maps f,g: X → Y between metric spaces are coarsely equivalent if d(f(x),g(x)) ≤ C for some C ≥ 0.
• A function f : X → Y between metric spaces is large-scale Lipschitz if there exist C,D ≥ 0 such that d(f(x₁),f(x₂)) ≤ C ⋅ d(x₁,x₂) + D for all x₁,x₂ ∈ X.

The notion of coarse maps f : X → Y is a slight generalisation of large-scale Lipschitz. Metric spaces X,Y are called coarsely equivalent if there exist coarse maps f : X → Y and g: Y → X such that f ∘ g and g ∘ f are coarsely equivalent to the respective identities.

To illustrate the situation, consider the sets X = {0,1,2,…} and Y = {2n : n ≥ 0} and Z = [0,∞) as subspaces of ℝ. Then X and Z are coarsely equivalent (via the inclusion X ⊆ Z) even though X is discrete and Z is not. In contrast, Y is not coarsely equivalent to Z and hence X and Y are not coarsely equivalent even though they are topologically indistinguishable discrete metric spaces.

The coarse category of metric spaces with coarse maps modulo coarse equivalences has been the subject of intense research in the past decade.

Recently, the supervisors of this project developed a new invariant called “coarse homotopy groups” which is inspired by the classical construction of homotopy groups in algebraic topology and is designed to capture coarse geometric phenomena. For example, π_coarse⁰ is strong enough invariant to tell that the metric space X,Y above are not coarsely equivalent.

The aim of this project is to study coarse homotopy groups, establishing both the theory and a collection of tools to facilitate explicit calculations. Prominent objectives are

• Find a useful notion of “coarse fibre bundles” and the associated long exact sequence in coarse homotopy groups.
• Formulate and prove theorems analogous to Seifert-Van Kampen theorem.
• Construct the coarse analogues of Eilenberg-MacLane spaces.
• Introduce and study Whitehead products in the setting of coarse homotopy groups.

Once set up, the machinery of coarse homotopy groups will be used to obtain fundamental results about groups equipped with metrics invariant to both left and right translations. This project will focus on the bi-invariant word metric on free groups with respect to their canonical set of generators. The student will work on, but not restricted to, the questions below.

• Free groups of ranks m and n are coarsely equivalent if and only if m = n.
• Calculate the coarse homotopy groups of the free groups. We conjecture that free groups are coarsely weakly connected.
• Study the asymptotic cones of free groups of finite rank. Decide whether they are contractible topological spaces.

Informal enquiries can be made by contacting Dr A Libman (a.libman@abdn.ac.uk).

Decisions will be based on academic merit. The successful applicant should have, or expect to obtain, a UK Honours Degree at 2.1 (or equivalent) in Pure Mathematics. Potential candidates The student must have basic knowledge in topology and group theory shown in their transcript.

We encourage applications from all backgrounds and communities, and are committed to having a diverse, inclusive team.

Application Procedure:

Formal applications can be completed online: https://www.abdn.ac.uk/pgap/login.php.

You should apply for Degree of Doctor of Philosophy in Mathematics to ensure your application is passed to the correct team for processing.

Please clearly note the name of the lead supervisor and project titleon the application form. If you do not include these details, it may not be considered for the project.

Your application must include: A personal statement, an up-to-date copy of your academic CV, and clear copies of your educational certificates and transcripts .

Please note: you do not need to provide a research proposal with this application.

If you require any additional assistance in submitting your application or have any queries about the application process, please don't hesitate to contact us at researchadmissions@abdn.ac.uk