International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract:

Affine manifolds are manifolds that admit an atlas with locally affine transition functions. A long-standing conjecture by Chern predicts the vanishing of the Euler characteristic for closed affine manifolds. For these manifolds the Euler characteristic is dominated by the simplicial volume, leading Bucher—Connell—Lafont to ask whether the simplicial volume of closed affine manifolds vanishes. In this talk I will show that certain complete affine manifolds have vanishing simplicial volume. The approach will be to study the simplicial volume of injective Seifert fiber spaces, manifolds that extend the notion of Seifert fibered 3-manifolds in higher dimensions. Our result also answers a question by Lück on the simplicial volume of aspherical manifolds in this setting. This is based on a joint work with Marco Moraschini.

Slides

Abstract:

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume. This is a joint work with Chris Connell and Shi Wang (arXiv:2410.19981). 

The first talk will mainly focus on some recent results in proving positivity of simplicial volume. I will outline the proof of our result in the second talk.

Slide 1, Slide 2

Abstract:

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume. This is a joint work with Chris Connell and Shi Wang (arXiv:2410.19981). 

The first talk will mainly focus on some recent results in proving positivity of simplicial volume. I will outline the proof of our result in the second talk.

Slide 1, Slide 2, Slide 3, Slide 4, Slide 5

Abstract:

We present a family of groups of  automorphisms of a regular tree that have almost prescribed local action (APLA) on the edges around the vertices. Since their introduction by Le Boudec, these groups have provided examples for addressing various group-theoretic questions.

In this talk, we prove a condition for the vanishing of their continuous bounded cohomology. Moreover, we show that when this condition is not satisfied, the continuous bounded cohomology in degree two is infinite-dimensional.

Slides

Abstract:

Let G be a countable non-elementary group acting properly on a metric space with contracting elements and H be a subgroup in G. We prove that H has infinite index in G if and only if the relative second bounded cohomology is infinite-dimensional. Our results generalize a theorem of Pagliantini-Rolli for finite-rank free groups. This is joint work with Renxing Wan.

Slides

Abstract:

Delta complexity and integral simplicial volume are two integral invariants associated with closed oriented manifolds. Although defined differently, these invariants share several properties, coincide in dimensions 1 and 2, and often exhibit similar behavior in higher dimensions as well.

In this talk, we will introduce both invariants and present a method to study them from an asymptotic perspective. This approach will lead us to construct a sequence of 3-manifolds for which the integral simplicial volume and delta complexity display distinct asymptotic behaviors.

This is joint work with Roberto Frigerio. 

Abstract:

Bounded cohomology is a powerful albeit very hard to compute invariant. Nothing encapsulates that more than the as of yet mysterious bounded cohomology of free groups. During this talk I will give a very brief introduction to bounded cohomology, further motivate why one should care about the bounded cohomology of free groups and then explain how to show that it is non-zero in degrees two, three and four.

Slides

Abstract:

Singular foliated simplices appeared first in Gromov’ work and were formalized by Sauer 20 years ago. We show how they can be organized into a suitable homology theory that allows to recover simplicial volume of manifolds. 

Motivated by the fruitful partnership between bounded cohomology and simplicial volume, we then focus on the dual theory. Via homological algebra we can connect foliated bounded cohomology with the theory of bounded cohomology of p.m.p actions recently settled with Savini. This provides new vanishing criteria for simplicial volume, based also on recent progresses about transverse groupoid with Hartnick.

Slides

Abstract:

I will talk about recent joint work with Nansen Petrosyan where we studied the behavior of L^2-Betti numbers under group-theoretic Dehn filling, a quotienting process of groups motivated by 3-manifold theory. As applications, we

verified the Singer Conjecture for Einstein manifolds constructed from arithmetic lattices of SO(n; 1). Another application appears in my collaboration with Francesco Fournier-Facio where we constructed the first uncountable family of finitely generated torsion-free groups which are mutually non-measure equivalent.

Slides

Abstract:

In this talk, we will introduce several coarse topological invariants of metric spaces, inspired by and analogous to, classical concepts in the study of group cohomology. Using the notion of an R-module over a metric space, we can interpret group cohomological notions, such as finiteness properties and cohomological dimension, for arbitrary metric spaces. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. Time permitting, we will discuss a higher dimensional analog of classical the Hopf-Freudenthal theorem that a group has either 0, 1, 2 or infinitely many ends.

Slides

Abstract:

In this talk we will revise the basic notion of the theory of simplicial sets and explain the definition of bounded cohomology of simplicial sets with local coefficients. This will allow us to state a version of the Serre spectral sequence in bounded cohomology. As an application at the end of the talk we will show how to deduce from the Serre spectral sequence in bounded cohomology a generalised version of Gromov's Mapping Theorem. This is a joint work with K. Li and G. Raptis.