Heegaard genus and Property 'tau' for hyperbolic 3-manifolds

Abstract

We 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 $\tau$. 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.