International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract:

In recent years, a clear relationship between approximate subgroups - subsets of subgroups stable under mutliplication up to a finite error relevant in many problems of combinatorial and geometric origin -  and bounded cohomology has emerged. In short, bounded cohomology can help measure the regularity of a given approximate subgroup. I will discuss recent developments in that direction by focusing on one example where that relationship can be exploited with great success. Namely, I will talk about approximate lattices, a rich class of approximate subgroups generalising lattices of locally compact groups that arises naturally in relations with aperiodic tilings, invariant point processes or Pisot numbers. 

Abstract:

The Charney-Davis strict hyperbolization is a procedure that turns polyhedra into spaces of negative curvature, while preserving some topological features. It has been used to construct examples of manifolds that exhibit unexpected features, despite having negative curvature. One may expect the fundamental groups of these manifolds to display strange features as well. On the other hand, we show that these groups admit nice actions on CAT(0) cube complexes, both in the absolute and relative settings. As an application, we obtain new examples of negatively curved Riemannian manifolds whose fundamental groups are virtually special and algebraically fibered.

Abstract:

One of the most celebrated results in the theory of bounded cohomology is Gromov's Mapping Theorem, which shows how the bounded cohomology of a topological space only depends on its fundamental group. Gromov's approach to the Mapping Theorem is based on the theory of multicomplexes, simplicial structures that lie in between simplicial complexes and simplicial sets.

The aim of this talk is to discuss how the theory of multicomplexes can be used to address a relative version of the Mapping Theorem. I will discuss the role of higher homotopy in relative bounded cohomology and some applications to the additivity of simplicial volume for aspherical manifolds.

Abstract:

The integral foliated simplicial volume of a manifold provides a dynamical version of its simplicial volume, measuring the "size" of fundamental cycles parametrised by a standard probability action of the fundamental group. Moreover, it gives a sharper upper bound for the L2-Betti numbers and for the cost of the fundamental group. We establish an integral formula for the parametrised simplicial volume along an ergodic decomposition of the underlying standard probability action.

Abstract:

 Let G be a semi simple Lie group with finite center and finitely many connected components and H a closed subgroups. When H is amenable it is well known that the continuous or measurable bounded cohomology of G can be obtained from the G-invariant cochains on G/H. We will see that this is in general no longer true for the usual (unbounded) cohomology of G. 

Specialising to the case when H=P is a minimal parabolic subgroup the difference between the cohomology of G and that of G/P (i.e. the kernel of the evaluation map) naturally decomposes as a direct sum of alternating and non-alternating classes. In particular, we will see that there exists cocycles on G/P, whose symmetrisation vanishes, representing non-trivial cohomology classes on G/P.

Time allowing, some applications to bounded cohomology will be discussed.

Joint work with Alessio Savini. 

Abstract:

We introduce the notions of asymptotic rank and injective hulls before investigating a coarse version of Dress’ 2(n+1)-inequality characterising metric spaces of combinatorial dimension at most n. This condition, referred to as (n,δ)-hyperbolicity, reduces to Gromov's quadruple definition of δ-hyperbolicity for n=1. The ℓ∞ product of n δ-hyperbolic spaces is (n,δ)-hyperbolic and, without further assumptions, any (n,δ)-hyperbolic space admits a slim (n+1)-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. Using tools from recent developments in geometric group theory, we look at some examples and show that every Helly group and every hierarchically hyperbolic space of asymptotic rank n acts geometrically on some (n,δ)-hyperbolic space. Joint work with Urs Lang.