Summer Semester 2021

Abstract:

The talk will be devoted to a new approach to the main theorem of the bounded cohomology theory, known as the Mapping Theorem, based on the ideas of Eilenberg-MacLane, Eilenberg-Zilber, and, especially, of Postnikov from 1950is. This approach allows to extend the Mapping Theorem to simplicial sets having the Kan extension property and also leads to a natural interpretation of the topological part of speaker's first proof (1984-5) of the Mapping theorem.

Abstract:

For a topological group G, a quasi-morphism on the universal covering of G is said to descend if it comes from a quasi-morphism on G. In this talk, we study the space of non-descending quasi-morphisms. As its application, we prove (un)boundedness of some characteristic classes on the group of (contact) Hamiltonian diffeomorphisms. This is a joint work with Shuhei Maruyama.

If time permits, we also explain our recent study on the space of non-extendable quasimorhisms, which is a joint work with Mitsuaki Kimura, Takahiro Matsushita, Shuhei Maruyama and Masato Mimura.

Abstract:

Several results in Bounded Cohomology theory are expressed by means of the multiplicity of amenable open covers. By studying the minimal cardinality of these covers one gets the notion of amenable category. topological complexity is another categorical invariant introduced by Farber in the context of topological robotics. In this talk we study the relation between amenable category and topological complexity.

Time permitting, we also discuss the monotonicity problem for degree-one maps and amenable category.

This is a joint work with Clara Löh and Marco Moraschini.

Abstract:

In this talk we will explicitly construct for free products of groups A*B quasimorphisms that happen to be invariant under all automorphisms of A*B. We will obtain Aut-invariant quasimorphisms on various groups of geometric origin ranging from the braid group on three strands over fundamental groups of knot complements to some right angled Coxeter groups. As a further application we will discuss the non-triviality of an Aut-invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura.

Abstract:

Despite its wide range of applications, bounded cohomology is hard to compute in general. In fact, it is an open problem whether the bounded cohomology of the non-abelian free group with trivial coefficients vanishes in higher degrees.

In this talk, we discuss an approach to this problem using cup products and explain how to use Delta-decompositions (due to Heuer) and aligned cochains (due to Bucher and Monod), to generate examples of trivial cup products in bounded cohomology of arbitrary degree greater than 3.

This is joint work with Michelle Bucher.

Abstract:

We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of l²-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. Group actions on Cantor spaces surprisingly play an important role in the proof. The talk is based on joint work with Sabine Braun.

Abstract:

Many interesting groups are generated by torsion elements, for instance, mapping class groups, SL(n, Z) and Homeo⁺(S¹). The word length with respect to this typically infinite generating set is called the torsion length. That is, the torsion length tl(gⁿ)/n of an element g is the smallest k such that g is the product of k torsion elements. The stable torsion length stl(g) is the limit of tl(gⁿ)/n, which measures the growth of the torsion length. I will explain how this is related to the stable commutator length and how to use topological methods to compute stl(g) in free products of finite abelian groups. The nature of the method implies that stl(g) is always rational in these free products. This is joint work with Chloe Avery.