The simplicial volume of hyperbolic manifolds with geodesic boundary

Abstract

Let n 3, let M be an orientable complete finite-volume hyperbolic n-manifold with compact (possibly empty) geodesic boundary, and let Vol.M / and kM k be the Riemannian volume and the simplicial volume of M . A celebrated result by Gromov and Thurston states that if @M D ∅ then Vol.M /=kM k D 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 @M ¤ ∅ then Vol.M /=kM k < v n .We prove here that for every Á > 0 there exists k > 0 (only depending on Á and n)As a consequence we show that for every Á > 0 there exists a compact orientable hyperbolic n-manifold M with nonempty geodesic boundary such that Vol.M /=kM k v n Á.Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for noncompact finite-volume hyperbolic n-manifolds without geodesic boundary. 53C23; 57N16, 57N65 Preliminaries and statementsLet X be a topological space, let Y Â X be a (possibly empty) subspace of X , and let R be a ring (in the present paper only the cases R D R and R D Z will be considered). For i 2 N we denote by C i .X I R/ the module of singular i -chains over R, ie the R-module freely generated by the set S i .X / of singular i -simplices with values in X . The natural inclusion of Y in X induces an inclusion of C i .Y I R/ into C i .X I R/, so it makes sense to define C i .X; Y I R/ as the quotient spaceThe homology of the resulting complex is the usual relative singular homology of the topological pair .X; Y / and will be denoted by H .X; Y I R/.

Stable complexity and simplicial volume of manifolds

“…We denote by v n the volume of the ideal regular hyperbolic simplex in H n . The following result is due to Thurston [46] and Gromov [19] (detailed proofs can be found in [3,7,43] for the closed case, and in [5,12,15,16] for the cusped case).…”

The simplicial volume of 3-manifolds with boundary

“…Several vanishing and nonvanishing results for the simplicial volume are available by now, but the exact value of nonvanishing simplicial volumes is known only in a very few cases. If M is (the natural compactification of) a complete finite-volume hyperbolic n-manifold without boundary, then a celebrated result by Gromov and Thurston implies that the simplicial volume of M is equal to the Riemannian volume of M divided by the volume v n of the regular ideal geodesic n-simplex in hyperbolic space (see [8,19] for the compact case and, for example, [2,4,6,7] for the cusped case). The only other exact computation of nonvanishing simplicial volume is for the product of two closed hyperbolic surfaces or more generally manifolds locally isometric to the product of two hyperbolic planes [3].…”

Integral foliated simplicial volume of hyperbolic 3-manifolds

“…The 5-chain link complement has a hyperbolic structure [26] and admits an ideal triangulation with 10 ideal and regular tetrahedra such that each edge has vertices in different cusps [25,Section 5.2]. By the proportionality between simplicial volume and Riemannian volume (which holds both in the compact case [13,31] and in the cusped case [7,10,11,5]) we have…”

“…In the present article, we will prove the following: Here, v 3 denotes the maximal volume of ideal geodesic 3-simplices in H 3 . The second equality in Theorem 1.1 is the classic proportionality principle for hyperbolic manifolds of Gromov [13] and Thurston [31], which also holds in more generality [7,10,11,5,29,4,9,1,19].…”