Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of non-compact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of non-compact locally symmetric spaces with finite volume of Q-rank at least 3 is zero.Comment: 33 pages; corrected the vanishing result (and adapted Section 5 accordingly), minor expository changes in the introductio

The spectrum of simplicial volume with fixed fundamental group

We study the spectrum of simplicial volume for closed manifolds with fixed fundamental group and relate the gap problem to rationality questions in bounded (co)homology. In particular, we show that in many cases this spectrum has a gap at zero. For such groups, this leads to corresponding gap results for the minimal volume entropy semi-norm and for the minimal volume entropy in dimension 4

Which finitely generated Abelian groups admit isomorphic Cayley graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.Comment: 16 pages; v2: added reference, reformulated quasi-convexity, v3: small corrections; to appear in Geometriae Dedicat

Integral foliated simplicial volume of hyperbolic 3-manifolds

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical manifolds and give refined upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is complemented by the calculation of the integral foliated simplicial volume of Seifert 3-manifolds

The spectrum of simplicial volume

Funder: University of CambridgeAbstract: New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in R≥0. In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the l1-semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation

BOUNDED COHOMOLOGY AND BINATE GROUPS

A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first nonamenable examples are the group of compactly supported homeomorphisms of Rn (Matsumoto–Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups F, T, and V is as simple as possible

Stable integral simplicial volume of 3‐manifolds

We show that non‐elliptic prime 3‐manifolds satisfy integral approximation for the simplicial volume, that is, that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory

ISOMORPHISMS IN ℓ¹-HOMOLOGY

Taking the ℓ¹-completion and the topological dual of the singular chain complex gives rise to ℓ¹-homology and bounded cohomology respectively. In contrast to ℓ¹-homology, major structural properties of bounded cohomology are well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to ℓ¹-homology. For example, we obtain a new proof of the fact that ℓ¹-homology depends only on the fundamental group and that ℓ¹-homology with twisted coefficients admits a description in terms of projective resolutions. The latter one in particular fills a gap in Park’s approach. In the second part, we demonstrate how ℓ¹-homology can be used to get a better understanding of simplicial volume of non-compact manifolds