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.

The simplicial volume of hyperbolic manifolds with geodesic boundary

“…A different result holds for hyperbolic manifolds with non-empty geodesic boundary: Kue03]). Let M be a hyperbolic n-manifold with non-empty geodesic boundary.…”

Generalizations of Agol’s inequality and nonexistence of tight laminations

“…We remark that Theorem 1 is not true without assuming amenability of π 1 ∂ 0 Q. Counterexamples can be found, for example, using [20] or [21], Theorem 6.3.…”

Geometrisation of 3-manifolds

“…This is a key tool in the proof of Theorem 13.1.3. In particular, we need an additivity result for the simplicial volume under gluing along incompressible tori (see [Gro82], [Kue03], [Som81]) which implies that the simplicial volume of a 3-manifold admitting a geometric decomposition is proportional to the sum of the volumes of the hyperbolic pieces. In particular, such a manifold has zero simplicial volume if and only if it is a graph manifold.…”

“…The following result is useful for computing the simplicial volume of a 3-manifold from its geometric decomposition [Gro82], [Kue03], [Som81], provided it exists. …”