Narrow Results

In this paper, we show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type, giving a converse to a result by Lafont and Schmidt within the scope of closed, locally homogeneous Riemannian manifolds. This characterizes all closed locally homogeneous Riemannian manifolds with nonzero simplicial volume.

In this paper, we provide new vanishing and glueing results for relative simplicial volume, following up on two current themes in bounded cohomology: The passage from amenable groups to boundedly acyclic groups and the use of equivariant topology. More precisely, we consider equivariant nerve pairs and relative classifying spaces for families of subgroups. Typically, we apply this to uniformly boundedly acyclic families of subgroups. Our methods also lead to vanishing results for ${\ell }^2$-Betti numbers of aspherical CW-pairs with small relative amenable category and to a relative version of a result by Dranishnikov and Rudyak concerning mapping degrees and the inheritance of freeness of fundamental groups.

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups, we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X, Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X, Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov's Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along π₁-injective boundary components with amenable fundamental group.

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.

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above.

Let $X$ be a topological space admitting an amenable cover of multiplicity $k\in \mathbb{N}$. We show that, for every $n\ge k$ and every $\alpha \in H_n(X;ℝ)$, the image of $\alpha$ in the ${\ell }^1$-homology module $H_n^{{\ell }^1}(X;ℝ)$ vanishes. This strengthens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the ${\ell }^1$-seminorm of $\alpha$ vanishes.