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 many 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 $PU(n,1)$ admits a finite index subgroup, which maps onto the integers with kernel of type $F_{n-1}$ and not $F_n$. 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.