Efficient fundamental cycles of cusped hyperbolic manifolds

Abstract

Let M be a manifold (with boundary) of dimension > 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the simplicial volume ∥M, ∂M∥, using ergodic theory of unipotent actions. As applications, we extend results of Jungreis and Calegari from closed hyperbolic to finite-volume hyperbolic manifolds: a) Strict subadditivity of simplicial volume with respect to isometric glueing along geodesic surfaces, and b) nontriviality of the foliated Gromov norm for most foliations with two-sided branching.