A quantitative version of a theorem by Jungreis

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume |M| of M is equal to vol(M)/v_n, where v_n is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio vol(M)/|M| is strictly smaller than v_n if M is compact with non-empty geodesic boundary. We prove here a quantitative version of Jungreis' result for n>3, which bounds from below the ratio |M|/vol(M) in terms of the ratio between the volume of the boundary of M and the volume of M. As a consequence, we show that a sequence {M_i} of compact hyperbolic n-manifolds with geodesic boundary is such that the limit of vol(M_i)/|M_i| equals v_n if and only if the volume of the boundary of M_i grows sublinearly with respect to the volume of the boundary of M_i. We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension three.