The simplicial volume of hyperbolic manifolds with geodesic boundary

TL;DR

This paper investigates the relationship between volume and simplicial volume of hyperbolic manifolds with geodesic boundary, establishing bounds and providing new proofs for proportionality principles in hyperbolic geometry.

Contribution

It proves a new bound relating boundary volume ratio to the volume-to-simplicial-volume ratio, and constructs manifolds approaching the ideal ratio, extending known proportionality results.

Findings

Established bounds linking boundary volume ratio to volume-to-simplicial-volume ratio.

Constructed hyperbolic manifolds with boundary approaching the ideal volume ratio.

Provided a new proof of the proportionality principle for hyperbolic manifolds.

Abstract

Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending on a and n) such that if the ratio between the volume of the boundary of M and the volume of M is less than k, then the ratio between vol(M) and ||M|| is greater than v_n-a. As a consequence we show that for every a>0 there exists a compact orientable hyperbolic n-manifold M with non-empty geodesic boundary such that the ratio between vol(M) and ||M|| is greater than v_n-a. Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for non-compact finite-volume hyperbolic n-manifolds without boundary.