2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/137188We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.Geometric TopologyGroup Theorytext