Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < PU(m, 1)$ of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to $\mathbb{Z}$ with kernel of type $\mathcal{F}_{m−1}$ but not of type Fm. This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer’s conjecture for aspherical Kähler manifolds.