Uniform waist inequalities in codimension two for manifolds with Kazhdan fundamental group

Let M be a closed Riemannian manifold with Kazhdan fundamental group. It is well known that the Cheeger inequality yields a uniform waist inequality in codimension one for the finite covers of M. We show that the finite covers of M also satisfy a uniform waist inequality in codimension two.
Comment: updated introduction; 14 pages