Hyperbolic groups form an important class of finitely generated groups that has\u00a0attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$\u00a0if it has a classifying space with finitely man cells of dimension at most $n$, generalising\u00a0finite presentability, which is equivalent to type $F_2$. Hyperbolic groups are of type $F_n$ for\u00a0all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties.\u00a0We use methods from complex geometry to show that every uniform arithmetic lattice with\u00a0positive first Betti number in $PU(n,1)$ admits a finite index subgroup, which maps onto\u00a0the integers with kernel of type $F_{n\u22121}$ and not $F_n$. This answers an old question of Brady\u00a0and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This\u00a0is joint work with Pierre Py.<\/p>\n","protected":false},"excerpt":{"rendered":"
Hyperbolic groups form an important class of finitely generated groups that has\u00a0attracted much attention in Geometric Group Theory. We call…<\/p>\n