Hyperbolic groups and their subgroups

Giles Gardam (University of Oxford)

Wednesday 15th February, 2017 16:00-17:00 Maths 515


Hyperbolic groups, introduced by Gromov in the 1980's, are "negatively curved" in a coarse sense. They generalize classes of groups such as small cancellation groups and fundamental groups of closed hyperbolic manifolds, but maintain many nice algebraic, algorithmic and topological properties. For instance, a torsion-free hyperbolic group can be realized as the fundamental group of a finite CW-complex with contractible universal cover. However, the subgroups of a hyperbolic group do not enjoy the same topological finiteness properties. In this talk I will outline a construction of R. Kropholler, building on work of Brady and Lodha, which gives an infinite family of hyperbolic groups with finitely presented subgroups which are non-hyperbolic by virtue of their lack of finiteness properties. We will conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.

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