Simplicial volume of compact manifolds with amenable boundary

Abstract

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.

On the equivalence of the definitions of volume of representations

“…Suppose that each component of ∂M has amenable fundamental group. In that case, it is proved in [6,23] that the natural inclusion i : (M , ∅) → (M , ∂M ) induces an isometric isomorphism in bounded cohomology,…”

Isometric embeddings in bounded cohomology

“…The amenability of π 1 (Y ) insures immediately, using the long exact sequence in relative bounded cohomology, the isomorphism of H n b (X, Y ) and H n b (X), but the fact that this isomorphism is isometric is, to our knowledge, not contained in Gromov's paper and requires a proof. This result was obtained independently by Kim and Kuessner [28], using the rather technical theory of multicomplexes. Our proof of Theorem 2 uses instead in a crucial way the construction of an amenable π 1 (X)-space thought of as a discrete approximation of the pair ( X, p −1 (Y )), where p : X → X is a universal covering.…”

Ideal Simplicial Volume of Manifolds with Boundary

“…Via some elementary duality results, this implies in turn that amenable spaces are somewhat invisible when considering their simplicial volume: for example, closed manifolds with amenable fundamental group have vanishing simplicial volume, and (under some mild additional hypothesis) the simplicial volume of manifolds with boundary is additive with respect to gluings along boundary components with amenable fundamental group. In Section 4 we introduce the dual theory to marked homology and, building on results from [BBF + 14,Fri17], we exploit duality to deduce the following: It is proved in [Löh07,KK15] that, under the assumptions of Theorem 4, the simplicial volume M of M coincides with the simplicial volume of the open manifold int(M ) = M \ ∂M (which is defined in terms of the locally finite homology of M \ ∂M [Gro82]), as well as with the Lipschitz simplicial volume of int(M ) (see [Gro82,LS09] for the definition). Therefore, for manifolds whose boundary components have amenable fundamental group, all these invariants also coincide with the ideal simplicial volume M I of M .…”