Finiteness properties of subgroups of hyperbolic groups

Venue:
Geb. 20.30, SR 2.058

Date:
13.07.2023

Speaker:
Claudio Llosa Isenrich

Time:
15:45 Uhr

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 F_{n} if it has a classifying space with finitely man cells of dimension at most n, generalising finite presentability, which is equivalent to type F_{2}.
Hyperbolic groups are of type F_{n} 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 F_{n1} and not F_{n}. This answers an old question of Brady and produces many finitely presented nonhyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.