International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract:

The goal of this talk is to present a method for proving bounded acyclicity (of some groups or complexes). The motivation for me was to prove the vanishing of bounded cohomology for p-adic groups, but I hope that the method can be useful in other contexts.

Abstract:

We give an overview of results proved in recent years that pertain to classifying the cobounded hyperbolic actions of any group. In other words, we discuss results that allow us to describe the partially ordered set of hyperbolic structures on a group. In certain cases, a complete classification of the poset is possible. In cases where this seems out of reach, we provide descriptions of large subsets of the poset. This covers a wide range of groups, including acylindrically hyperbolic groups, nilpotent groups, many solvable groups and ``Thompson-like" groups. 

Slides

Abstract:

By Gromov's Vanishing Theorem, the comparison map of a topological space vanishes in presence of amenable open covers of controlled multiplicity. This result can be approached from several points of view. My aim is to present Gromov's original approach via the theory of multicomplexes, whose foundations have been laid down by Frigerio and Moraschini. This perspective is helpful to obtain relative versions of Gromov's theorem, which are inspired by the recent work of Li, Löh and Moraschini.

Slides

Abstract:

A problem that attracted a lot of attention is understanding how wild can a subgroup H < G₁ x ... x G_n be, where the G_i are groups in a fixed class, for example free groups. The finiteness properties of H play an important role: by a result of Bridson, Howie, Miller, and Short, if the G_i are free (or surface) groups and H is of type F_n, then H must be virtually a direct product itself. If we lower our requirement on H to F_{n-1} we obtain examples like the Stallings-Bieri groups SB_n arising as a kernel of F₂ x ... x F₂ → ℤ. By a result of Kuckuck, all such H arise virtually as a kernel of a map to ℤ^r. 

The simplest example in this class is the group K³₂(2), defined as the kernel F₂ x F₂ x F₂ → ℤ². Its Dehn function was known to be bounded between n³ and n⁶; with a topological argument, we show that it is in fact quartic. This is joint work with Ascari, Bertolotti, Italiano and Llosa Isenrich.

Slides

Abstract:

We first discuss various embedding results for groups in the literature. Then we talk about how could one embed groups into boundedly acyclic groups using the labeled Thompson groups and twisted Brin Thompson groups. This is based on a joint work with Fan Wu, Mengfei Zhao and Zixiang Zhou.

Slides

Abstract:

For almost all compact surfaces, the existence of a homogeneous quasi-morphism on the group of homeomorphisms which are isotopic to the identity has been proven using the fact that these groups act on hyperbolic spaces, namely fine curve graphs. Using a recent result of Bowden, Hensel and Webb, it is now possible to prove the existence of a homogeneous quasi-morphism in the case of the Klein bottle.

Slides

Abstract:

We will provide an overview of different types of quasimorphisms and existing vanishing results for cup products and Massey products with their induced classes in (equivariant) bounded cohomology. Then we discuss how to generalise these quasimorphisms to reprove and extend the existing vanishing results in a unified way.

Slides

Abstract:

Some of the classical results about the finiteness properties of discrete groups can be transferred to the case of totally  disconnected locally compact groups. In this talk I will give a short introduction to the finiteness properties of discrete and tdlc groups. Then I will introduce the Schlichting completion, with which it is possible to construct a tdlc group from a discrete group. After that I will give an idea how we can transport some finiteness properties along these construction. 

Abstract:

I will discuss an $\ell^\infty$-cohomology characterisation of hyperbolicity for general metric spaces, and a related characterisation of acylindrically hyperbolic groups.

Slides 

Abstract:

40 years ago, Michael Davis found a beautiful and powerful method for constructing aspherical manifolds, which since then has been used many times to find examples of aspherical manifolds with interesting properties. I will talk about a strategy to study the simplicial volume of these manifolds. Can we characterize the ones having positive simplicial volume? Do they satisfy Gromov's conjecture about the relation between simplicial volume and Euler characteristic? The approach is based on the study of triangulations of spheres (which is the data needed by Davis' method to produce manifolds) and simplicial maps between them.

Abstract:

I will describe two new constructions of finitely presented infinite simple groups. Among these are the first examples of finitely presented simple left orderable groups, and a sequence of uniformly simple groups whose Ulam width is increasing.