International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract:

There's a conjecture that a compact Riemannian manifold with almost nonnegative scalar curvature, with respect to a normalized volume, has vanishing simplicial volume.  I'll explain the background to this conjecture, how it motivates problems about the existence of finite volume Riemannian metrics with positive scalar curvature on noncompact manifolds, and some results in the spin case.

Abstract:

Stable commutator length is a measure of homological complexity of group elements, which is known to take large values in the presence of various notions of negative curvature. We will present a new geometric proof of a theorem of Heuer on sharp lower bounds for scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl.

Abstract:

The space of subgroups of a countable group G is a compact topological space equipped with a continuous action of G which encodes many of the properties of its non-free actions. I will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees, and give some conditions under which the conjugation action on the perfect kernel of this space is highly topologically transitive. I will then explain how this information can be used to find many new examples of groups which admit a faithful transitive amenable action, including for instance all virtually compact special groups.

This is joint work with Damien Gaboriau.

Abstract: 

In 1996 Gersten proved that if G is a word hyperbolic group of cohomological dimension 2 and H is a subgroup of type FP_2, then H is hyperbolic as well. In this talk, I will present a project with Robert Kropholler and Vlad Vankov generalising this result to show that the same is true if G is only assumed to have cohomological dimension 2 over some ring R and H is of type FP_2(R) .

Abstract: 

Milnor in 1958 proved that an orientable plane bundle E over a surface of genus g admits a flat connection if and only if the absolute value of the Euler number of E is bounded by g-1. Later Wood in 1971 generalized this result to flat circle bundles over surfaces. Ghys in 80s asked if Wood’s inequality can be generalized to flat oriented S³-bundles. In a joint work with Monod, we showed that the first Pontryagin class and the Euler class are in fact unbounded for flat oriented S³-bundles. This is the first talk in a two-series talk that I will report on a joint work with Fournier-Facio and Monod in which we prove that topological Pontryagin classes for flat Euclidean bundles are all unbounded.

Abstract: 

Last week we heard about the bounded acyclicity of Homeo(R^n) and its consequences. In this talk, I will present the proof of this result, joint with Nicolas Monod and Sam Nariman. More generally, I will introduce a criterion for bounded acyclicity of transformation groups that are not compactly supported, which applies in many other contexts, and sometimes can even prove ordinary acyclicity.

Abstract: 

The classical Gambaudo-Ghys construction, that produces quasimorphisms on Diff_0(M,vol) out of quasimorphisms on \pi_1(M), has various descriptions and generalisations. The one I would like to explain in this talk takes a class in the cohomology of the mapping class group of M and returns a class in the cohomology of Homeo(M,vol) (no zero in the subscript). We will give two descriptions: geometric and in the language of couplings. This construction allows to create a Euler-class-like cohomology classes in the cohomology of Homeo(S,vol) for a closed surface S. This is joint work with Michael Brandenbursky.

Abstract: 

It is a classical result that manifolds that are total spaces of  fiber bundles, whose fiber has amenable fundamental group, have vanishing simplicial volume. In this talk I will explore the opposite route, where the manifolds fiber non-trivially with arbitrary fiber over a simply-connected manifold. While a general answer seems to be out of reach, I will explain how one obtains vanishing results for many connected structure groups.