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The University of Southampton
Mathematical Sciences

Research Group: Geometric Group Theory

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The central idea of Geometric Group Theory is to study discrete groups by viewing them as metric spaces, using for example a word metric. Gromov introduced this viewpoint, with his idea of hyperbolic groups and with his proof of a conjecture of Milnor. A key idea in Gromov's proof was that when `viewed from far away', a free abelian group starts to look like a real vector space.

Many different aspects of Geometric Group Theory are represented in the research of Southampton mathematicians: hyperbolic groups and their generalizations; automorphism groups of free groups and free products; CAT(0) groups and the important special case CAT(0) cubical groups; group actions on Hilbert spaces; Coxeter groups; Artin groups; Bestvina-Brady groups and generalizations; homological finiteness conditions; classifying spaces for groups and families; soluble groups and amenable groups; connections with manifolds and topology; Fuchsian and Kleinian groups; connections with non-commutative geometry.

If you are interested in working with us on any of these topics do not hesitate to get in touch.

Here is a sample of research topics that we work on:

  • Acylindrically hyperbolic groups: a generalization of Gromov's already very broad notion of a hyperbolic group. The general theory is still in its infancy, and there are many natural questions that can be asked.
  • Generalized Bestvina-Brady groups: Bestvina and Brady made a functor from finite simplicial complexes to groups. Their functor applied to a non-contractible complex with trivial homology gave rise to groups that are FP but not finitely presented. Generalized Bestvina-Brady groups depend on a finite complex plus a set of integers, so there are many more of them. Their homological properties are well understood now, but there are many open questions concerning these groups.
  • Coxeter groups: infinite Coxeter groups have many applications. Davis's trick involving these groups enables one to create Poincare duality groups with very wild properties. They also give lots of interesting examples for classifying spaces for families.

The group welcomes applications for postgraduate studies. Please contact group members (email, telephone, or in person) for more information; and when ready, please apply through the Graduate School application page.

List of related projects to Geometric Group Theory
Related ProjectsStatus
Boundary representations of groupsActive
Profinite topology on non-positively curved groupsActive
Coarse geometry of non-positively curved spacesActive
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