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 Z with kernel of type Fm-1 but not of type F-m. 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 Kahler manifolds.