We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < { \rm PU}(m,1)$ of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to $\mathbb{ Z}$ with kernel of type $\mathscr{ F}_{ m-1}$ but not of type $\mathscr{ F}_{ m}$. 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 K\"ahler manifolds.
Categories: math.GR math.CV math.DG math.GT