A quantitative version of a theorem by Jungreis

2015/03/12 by Michelle Bucher, Bucher, Michelle, Roberto Frigerio +3
Mathematics · #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.DG #math.GT

paper · pdf · doi:10.48550/arxiv.1503.03837

2 figures, formerly part of arXiv:1208.0545

arxiv created 2015/03/12 · arxiv updated 2015/03/13

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.

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