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.
Although hyperbolicity of a group defined by a finite presentation is
undecidable in general, the programs in the author's KBMAG package, which is
implemented in GAP and in Magma, can verify hyperbolicity when the property
holds.
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. There is
a GAP implementation by Marcus Pfeiffer and also a recent Magma implementation
by the author.
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, and applied to
infinite families of group presentations.