A quantitative version of a theorem by Jungreis

TL;DR

This paper provides a quantitative relationship between the simplicial volume and volume of hyperbolic manifolds with boundary, extending Jungreis' theorem and analyzing the asymptotic behavior of sequences of such manifolds.

Contribution

It establishes a lower bound for the ratio of simplicial volume to volume based on boundary volume, and offers estimates for 3-dimensional cases, advancing understanding of hyperbolic manifolds with boundary.

Findings

Bounds the ratio |M|/vol(M) using boundary volume in higher dimensions.

Characterizes when the volume-to-simplicial volume ratio approaches v_n.

Provides estimates for simplicial volume in 3-dimensional hyperbolic manifolds.

Abstract

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.