Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

Abstract We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < {\mathrm{PU}}(m,1)$ Γ < PU ( m , 1 ) of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to ℤ with kernel of type $\mathscr{F}_{m-1}$ F m − 1 but not of type $\mathscr{F}_{m}$ 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 Kähler manifolds.