A quantitative version of a theorem by Jungreis

作者：Bucher Michelle; Frigerio Roberto^*; Pagliantini Cristina

来源：Geometriae Dedicata, 2017, 187(1): 199-218.

DOI：10.1007/s10711-016-0197-6

• Simplicial volume
• Riemannian volume
• Hyperbolic manifold
• Geodesic boundary
• Pseudomanifold
• Geodesic simplices

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume of parallel to M parallel to of M is equal to Vol(M)/nu(n), where nu(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)/parallel to M parallel to is strictly smaller than nu(n) if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis' result for n >= 4, which bounds from below the ratio parallel to M parallel to/Vol (M) in terms of the ratio Vol(partial derivative M)/Vol(M). As a consequence, we show that, for , a sequence of compact hyperbolic n-manifolds with geodesic boundary satisfies lim(i) Vol(Mi())/parallel to M-i parallel to = nu(n) if and only if lim(i) Vol(partial derivative M-i)/Vol(M-i) = 0. We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.

• 出版日期2017-4