A quantitative version of a theorem by Jungreis

Abstract

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume $$\Vert M\Vert $$ of M is equal to $$\mathrm{Vol}(M)/v_n$$ , 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 $$\mathrm{Vol}(M)/\Vert M\Vert $$ is strictly smaller than $$v_n$$ if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for $$n\ge 4$$ , which bounds from below the ratio $$\Vert M\Vert /\mathrm{Vol}(M)$$ in terms of the ratio $$\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)$$ . As a consequence, we show that, for $$n\ge 4$$ , a sequence $$\{M_i\}$$ of compact hyperbolic n-manifolds with geodesic boundary satisfies $$\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n$$ if and only if $$\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0$$ . We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.