Summer Semester 2022
Program:
04/04/2022 - Hannah Alpert (Auburn University)
3-manifolds with small Urysohn 1-width
Abstract:
If every unit ball in a Riemannian n-manifold has small volume, then the manifold can be mapped to an (n-1)-complex with every fiber of the map having small diameter; this theorem has proofs by Guth in 2017 and Liokumovich--Lishak--Nabotovsky--Rotman and Papasoglu in 2019. It is a variant of a conjecture by Gromov from 1983, which replaces the volume bound on unit balls by a hypothesis that the manifold has positive scalar curvature, and replaces the (n-1)-complex in the conclusion by an (n-2)-complex. Is there another version that combines the hypothesis about small unit balls with the stronger conclusion about mapping to an (n-2)-complex? We state such a conjecture for all n, and prove a version of it for n=3. Joint work with Alexey Balitskiy and Larry Guth.
11/04/2022 - Vladimir Vankov (University of Southampton)
Large quasi-isometry classes via bounded cohomology
Abstract:
The study of quasi-isometries between finitely generated groups has traditionally been one of the more common questions of geometric group theory, which includes understanding the possible nature of quasi-isometry classes in general. We explore generalising constructions of uncountably many quasi-isometric groups to the torsion-free setting, by appealing to phenomena not present in the countable setting. The role of bounded cohomology is to establish quasi-isometries, using ideas dating back to Gersten.
25/04/2022 - Italian holidays
02/05/2022 - Erika Kuno (Osaka University)
Fine curve graphs and the Gromov hyperbolicity for nonorientable surfaces
Abstract:
A fine curve graph defined by Bowden, Hensel, and Webb is a new curve graph consists of the actual essential simple closed curves on a surface. They proved that the fine curve graph of any closed orientable surface of genus g≥1 is uniformly hyperbolic in the sense of Gromov.
In this talk, we introduce the framework by Bowden, Hensel, and Webb for orientable surfaces, and explain how to generalize their result to closed nonorientable surfaces. Especially we would like to talk about differences from the case of orientable surfaces.
This is a joint work with Mitsuaki Kimura.
09/05/2022 - Matthias Uschold (University of Regensburg)
L²-Betti numbers and computability of reals
Abstract:
L²-Betti numbers can be seen as an equivariant sibling of ordinary Betti numbers. They share many properties. However, L²-Betti are a priori only non-negative real numbers.
In this talk, I will report on recent work on compatibility aspects of these numbers, thus giving a characterisation of which numbers actually occur (given some hypotheses). This is joint work with Clara Löh.
We will also discuss some known results and conjectures about the relation between L²-Betti numbers and simplicial volume.
16/05/2022 - Domenico Marasco (University of Pisa)
Bounded cohomology via differential forms and cup product
Abstract:
Integrating over straight simplices defines a map from the space of closed differential k-forms of a negatively curved Riemannian manifold to its degree k bounded cohomology. In particular, in a 1988 paper J. Barge and E. Ghys showed that the case of closed surfaces S and k=2 is particularly interesting since this map is injective and thus Ω² (S) defines an infinite dimensional subspace of H²b(S).
We will have a look at some facts about bounded cohomology classes defined by differential forms. Then we will show that the cup product of a class defined by an exact 2-form with any other class is always trivial in bounded cohomology.
23/05/2022 - Kevin Li (University of Southampton)
A vanishing theorem for relative simplicial volume
Abstract:
One of the strongest vanishing results for simplicial volume of closed manifolds is in the presence of amenable covers. We establish a relative version for manifolds with π₁ -injective boundary. To do so, we follow the recent approach by Löh and Sauer via classifying spaces for families of subgroups.
This is joint work with Clara Löh and Marco Moraschini.
30/05/2022 - George Raptis (University of Regensburg)
The vanishing of the simplicial volume and the Euler characteristic
Abstract:
A well-known open question of Gromov asks whether the vanishing of the simplicial volume of an aspherical manifold implies the vanishing of its Euler characteristic. In this talk, I will discuss several approaches to relate these two fundamental invariants and present some recent partial results in connection with Gromov's question.
This is based on joint work with C. Löh and M. Moraschini
06/06/2022 - Federica Bertolotti (SNS Pisa)
Filling volume and simplicial volume of mapping tori.
Abstract:
Let M be a closed orientable n-manifold. The Real Filling Volume and the Integral Filling Volume are numerical invariants on the set of orientation preserving self-homotopy equivalences of M, which are defined in terms of filling norm on the space of boundaries of M.
We will study several properties of these two invariants and state some vanishing results.
We will investigate the (strong) relations between the real filling volume of an homeomorphism f and the simplicial volume of the mapping torus of f and the (less strong) relations between the integral filling volume of f and the stable integral simplicial volume.
Finally, we will see possible applications of these two invariants.
Email:
bounded.cohomology@gmail.com
Organizers:
Caterina Campagnolo (UAM Madrid)
Francesco Fournier-Facio (ETH Zurich)
Clara Löh (University of Regensburg)
Marco Moraschini (University of Bologna)