Summer Semester 2020

Abstract:

Grigorchuk's Density Theorem states that any alternating real-valued quasimorphism of a free group may be expressed as an infinite sum of Brooks quasimorphisms. It is not clear which sums give rise to actual quasimorphisms, something which was already discussed in Grigorchuk's paper.

However, when we restrict to Brooks quasimorphism on non-self-overlapping words, the question becomes more approachable, and such sums may be studied combinatorially. We will discuss these quasimorphisms, giving a partial result which suggests that they may be enough to describe the whole bounded cohomology of the free group. Time permitting, we will discuss some subfamilies that fall into Heuer's framework of decompositions, yielding new trivial cup products.

Abstract:

For a connected, oriented topological manifold, the Gromov norm associates each homology class a real value which measures how efficiently the class can be represented in terms of the linear combinations of singular simplices. Although the notion is defined topologically, it is closely related to the geometry of the manifold. In case of negatively curved manifolds, Gromov and Thurston uses the geodesic straightening to show the positivity of Gromov norm on degree 2 and above. For higher rank symmetric spaces, Lafont and Schmidt uses the barycentric straightening to show the positivity of the simplicial volume. In this talk, we will discuss the barycentric straightening and show how one can make use of the local estimate to show the positivity of the simplicial volume, thus extending the result to a large class of nonpositively curved manifolds.

Abstract:

Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications.

In dimensions two and three, the set of possible values of simplicial volume may be fully computed using geometrization, but is hardly understood in higher dimensions. In joint work with Clara Löh (University of Regensburg), we show that the set of simplicial volumes in higher dimensions is dense in the non-negative reals. I will also discuss how the exact set of simplicial volumes in dimension four or higher may look like.

Abstract:

We study the algebraic structure of three dimensional bounded cohomology generated by volume classes for infinite co-volume, finitely generated Kleinian groups. While bounded cohomology is generally unwieldy, we show that addition admits a natural geometric interpretation for the volume classes of tame hyperbolic manifolds of infinite volume and bounded geometry: the volume classes of singly degenerate manifolds sum to the volume classes for manifolds with many degenerate ends. It turns out that this generates the linear dependencies among certain geometric classes, giving a complete description of the algebraic structure of some geometrically defined subspaces of bounded cohomology.

We will indicate some problems left open by this discussion and give some suggestions for future directions. Definitions, background, and geometric aspects of hyperbolic manifolds homotopy equivalent to a closed surface will be introduced as needed during the talk. We will assume some familiarity with Gromov hyperbolic metric spaces.

Abstract:

The simplicial volume defined by Gromov measures the complexity of the fundamental class of a manifold based on the real numbers and the usual absolute value. It seems natural to study variations of simplicial volume, which are obtained by altering the coefficients and the underlying notion of "absolute value". This construction provides a rich family of invariants of manifolds. In this talk we introduce such variations of simplicial volume and, in particular, discuss p-adic simplicial volumes and some of their basic properties.

To provide an example we compute the p-adic simplicial volumes of surfaces using a combinatorial method.

Our understanding of p-adic simplicial volumes is currently rather incomplete and we plan to mention a number of open problems.

All this is based on joint work with Clara Löh.

Abstract:

Central extensions of a given group G by, say, Z are in bijection with the second cohomology of G.

In light of this bijection, bounded cohomology has something to say about the geometry of a central extension, meaning that if the cohomology class associated to a central extension is bounded, then the extension is quasi-isometrically trivial, so that in particular it is quasi-isometric to a product.

However, it turns out that the converse does not hold, meaning that there are quasi-isometrically trivial extensions whose associated cohomology class is not bounded. I will discuss such an example, and I will also discuss a few large classes of groups where the converse does hold.

Joint work with Roberto Frigerio.

Abstract:

The theory of numerical invariants has been strongly exploited to prove rigidity behaviors of representations of lattices in the context of Hermitian Lie groups.

Recently the theory has been formulated in the wider context of measurable cocycles, and this allowed several adaptations of results proved for representations.

The aim of this talk is first to introduce the notion of numerical invariants for representations using maps between bounded cohomology groups. We will show how boundaries and boundary maps play a crucial role in the study of rigidity by presenting an example.

The second part will be dedicated to a brief introduction to measurable cocycles and to the definition of numerical invariants in this context. We will try to understand on the one hand what follows straightforwardly from the classical theory for representations, and on the other hand what are the nodus and the main differences.

Slides

Slides

Abstract:

I will discuss recent work with Sebastian Hensel and Richard Webb on building quasi-morphisms on the identity component of diffeomorphism of higher genus surfaces, thus answering a question posed by Burago-Polterovich-Ivanov in 2008. The main tool for this is a new "dagger" curve graph, which is a Gromov hyperbolic space on which surface diffeomorphisms naturally act. I aim to put the result into some context and outline the main ideas of the proof.

Abstract:

Stable integral simplicial volume is an example of a gradient invariant: It is the infimum of the normalised integral simplicial volumes of all finite coverings. Analogously to other gradient invariants (such as rank gradients or Betti number gradients) there is also a dynamical description available. This dynamical view can be beneficial in computations. In this talk, I will give a quick introduction to this viewpoint and its applications.