Computational Algebra Seminar: Holt -- Hyperbolicity of groups

Speaker:

Derek Holt (University of Warwick).

Title:

Computational techniques for proving that a group is hyperbolic.

Abstract:

A finitely presented group is called hyperbolic if geodesic triangles in
its Cayley graph are uniformly thin or, equivalently, if its Dehn function
is linear.

The programs in the author’s KBMAG package, which is implemented in Magma,
can verify hyperbolicity of a given finitely presented group.

In this talk we describe new methods for proving hyperbolicity and for
estimating the Dehn function that are based on small cancellation theory
and the analysis of the curvature of van Kampen diagrams for the group.
The first version of a Magma implementation is available.

These methods are due to Richard Parker and many others. They have the
disadvantage that they are not guaranteed to succeed on every hyperbolic
group presentation, but when they do they are generally much faster than
KBMAG. They can also sometimes be carried out by hand on infinite families
of presentations, whereas KBMAG can only be applied to individual
presentations.