Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on M whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

Abstract. We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds.This paper is devoted to the behaviour of simplicial volume under cutting and pasting along amenable submanifolds. Such results have been proved by Gromov in [3], for closed manifolds ([3], Section 3) by fairly elementary arguments, and for open manifolds ([3], Section 4) by much more advanced arguments. The aim of this article is to write complete proofs for the case of compact manifolds with boundary, using elementary arguments which are closer to the arguments that were needed for the closed case.The simplicial volume is a homotopy invariant of compact manifolds. It was defined in [3], using the l 1 -norm on singular chains, as follows.

For a closed locally symmetric space M = Γ\G/K and a representation ρ : G → GL (N, C) we consider the push-forward of the fundamental class in H * BGL Q and a related invariant in K * Q ⊗ Q. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric spaces of R-rank one.(C) is, up to torsion, isomorphic to the Bloch *will be called the Eilenberg-MacLane map. The chain homotopy inverse is given by the composition of str with the chain isomorphism Φ, thusThe geometric realization | BΓ | of BΓ in the sense of [22] is aspherical by [20, p.128]. Given a manifold M and an isomorphism I : π 1 M ∼ = Γ, there is an up to homotopy unique continuous mapping h M : M →| BΓ | which induces I on π 1 , see [20, p.177]. The map h M (rather its homotopy class) is called the classifying map for π 1 M . If M is aspherical and has the homotopy type of a CW-complex then h M is a homotopy equivalence, and h M * : H * (M ; Z) → H * (| BΓ |; Z) is the composition of EM −1 with the isomorphism i * : H simp * (BΓ; Z) → H * (| BΓ |; Z) that is induced by the inclusion i of the simplicial into the singular chain complex.

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.

Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic Jaco-Shalen-Johannson-submanifold of M − S and show, in particular, that Guts(M, S) is empty.