Finiteness properties of subgroups of hyperbolic groups
Geb. 20.30, SR 2.058
Claudio Llosa Isenrich
Abstract: Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type Fn if it has a classifying space with finitely man cells of dimension at most n, generalising finite presentability, which is equivalent to type F2.
Hyperbolic groups are of type Fn for all n and it is natural to ask if their subgroups inherit these strong finiteness properties. We use methods from Complex Geometry to show that every uniform arithmetic lattice with positive first Betti number in PU(n,1) contains a subgroup of type Fn-1 and not Fn. This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.