Winter Semester 2020/2021

Abstract:

In his pioneering paper "Volume and bounded cohomology" Gromov introduced the notion of bounded cohomology of topological spaces. Among the foundational results, there is the well-celebrated Mapping Theorem, which shows how bounded cohomology only depends on the fundamental group of the given space.

The aim of this talk is to explain the ideas behind Gromov’s proof of the Mapping Theorem. One crucial ingredient is the theory of multicomplexes, a simplicial structure which lies in the middle between simplicial complexes and simplicial sets.

Most of the results discussed in the talk are jointly with Roberto Frigerio.

Abstract:

A classical striking result in hyperbolic geometry, known as Mostow-Prasad rigidity, states that n-hyperbolic closed manifolds with isomorphic fundamental group are in fact isometric.

A very nice proof of this fact for closed orientable manifolds, due to Gromov, makes strong use of simplicial volume. More precisely, a fundamental step in the proof is to show that hyperbolic volume and simplicial volume coincide up to a constant. The aim of this talk is to understand the skeleton of the proof of such a result, known as Gromov proportionally principle, proving formally the “easy” inequality and sketching the converse part which is a little more technical.

The complete proof can be found in the books by Thurston, Benedetti and Petronio or the more recent one by Martelli.

Abstract:

In this talk, I will introduce the barycentric straightening in the general context of non-positively curved manifolds. I will go through the derivative estimate which gives rise to the Jacobian estimate of the straightening map. If the Jacobian is uniformly bounded, then the volume of all straightened simplices have uniformly bounded volume, which will imply that the simplicial volume of the manifold is strictly positive. I will discuss in certain cases of non-positively curved manifolds when the Jacobian estimate can be obtained via an eigenvalue matching process. In particular, I will explain why this estimate works in SL_4(R) but not in SL_3(R).

Abstract:

Gromov’s systolic inequality relates the length of a shortest non-contractible loop to the volume of a Riemannian manifold.

Moreover, Gromov showed that the optimal constant appearing in the systolic inequality (systolic volume) depends on the topology of manifolds. For example, systolic volume is related to simplicial volume.

The aim of this talk is to briefly introduce the ideas of Gromov’s proof of the theorem relating systolic volume to simplicial volume. One of the main tools used in Gromov's proof is the smoothing technique, which relies on an alternative definition of simplicial volume. The smoothing technique was originally introduced by Gromov in the paper Volume and bounded cohomology.

Abstract:

A famous open question by Gromov asks the following: Take a closed connected oriented aspherical manifold. If its simplicial volume vanishes, does its Euler characteristic vanish as well?

Motivated by this question, we introduce different variants of the classical simplicial volume and explain how they can help tackle Gromov's question. We explore their relations with each other and with other topological invariants. We also present an overview of the current state of research around them.

Abstract:

We explain how the theory of group von Neumann algebras allows to define equivariant Betti-numbers for infinite group actions.

The so obtained L²-Betti numbers have proven to be powerful tools in various contexts, including simplicial volume and variants thereof.

Abstract:

See above (two lines, above).

Abstract:

As Nicolaus Heuer pointed out last week, the stable commutator lengths can be defined using quasimorphisms. We will now study quasimorphisms on groups from the viewpoint of group actions.

After a short introduction on hyperbolic groups and mapping class groups, we will give a classical criterion for showing the existence of nontrivial quasimorphisms on groups acting nicely on hyperbolic graphs following Bestvina-Fujiwara. We then finish our talk by checking that both hyperbolic groups and mapping class groups satisfy this interesting criterion.

Abstract:

In the last talk we saw how to construct non-trivial quasimorphisms on groups acting nicely on hyperbolic spaces, which implies that their second bounded cohomology is large. The construction was a more complicated version of the Brooks quasimorphisms, which are defined on the free group.

In this talk we will see another way to exploit the knowledge about the bounded cohomology of the free group to prove non-vanishing results for much more general groups, and even in higher degrees. Namely, we will explain how to extend quasicocycles (quasimorphisms and higher-dimensional analogues) from a subgroup to the ambient group, under the condition that the subgroup is hyperbolically embedded. This notion generalizes that of relative hyperbolicity, and applies to the larger class of acylindrically hyperbolic groups, proving that all such groups have large second and third bounded cohomology.

Abstract:

A symmetric space is a Riemannian manifold with isometric point reflections about every point. This class of spaces contains many of (at least) my favourite examples of Riemannian manifolds.

Nevertheless one can still classify symmetric spaces completely, exploiting their association with Lie groups. Getting back to the topic of the seminar, we present the definition of continuous and continuous bounded cohomology of a locally compact group. A famous open conjecture asks if these two notions coincide for semisimple connected Lie groups with finite center. We will discuss two resolutions of continuous (bounded) cohomology in this case and hopefully present some evidence for the conjecture.

Abstract:

Given a torsion-free lattice Γ ≤ G in a simple Lie group of non-compact type, one of the main problems in the theory of geometric structures is to understand the space of representations ρ : Γ → H into a locally compact group.

Surprisingly continuous bounded cohomology is a powerful tool in this kind of investigation. Indeed, under suitable hypothesis, one can exploit bounded cohomology techniques to define a numerical invariant associated to each representation. Such an invariant is called multiplicative constant and it usually has bounded absolute value.

Representations which maximize their multiplicative constant are called maximal and they define a nice subset of the representation space.

In this talk we are going to define the notion of multiplicative constant and to study some applications like the Euler invariant and the Volume invariant.

Abstract:

IIn Filippo Sarti's talk we have seen a geometric proof of Gromov's proportionality principle, a classical result that establishes a connection between the topology of a closed Riemannian manifold (namely, its simplicial volume) and its geometric structure.

In this chapter we will present a more algebraic proof, relying heavily on the isometric isomorphism between singular and continuous cohomology.

Our strategy involves finding a specific continuous cohomology class (the volume coclass), whose l^∞-seminorm will determine the proportionality constant between Riemannian and simplicial volume.

Finally, as an application, we will compute this constant for hyperbolic manifolds, showing that their simplicial volume is non-vanishing.