Constructions in stable commutator length and bounded cohomology

Thesis

The bounded cohomology of a group G with coefficients in a normed G-module V was first systematically studied by Gromov in 1982 in his seminal paper [Gro82] in connection to the minimal volume of manifolds. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for the most basic groups: There is no finitely generated group G for which the full bounded cohomology with real coefficients is known except where it is known to vanish in all degrees (see [Mon06]). In this thesis we discuss several new constructions for classses in bounded cohomology.

There is a well-known interpretation of ordinary group cohomology in degrees 2 and 3 in terms of group extensions. We establish an analogous interpretation in the context of bounded cohomology. This involves certain maps between arbitrary groups called quasihomomorphisms, which were defined and studied by Fujiwara and Kapovich in [FK16].

A key open problem is to compute the full bounded cohomology of a non-abelian free group F with trivial real coefficients. It is known that the bounded cohomology in dimension n is trivial for n = 1 and infinite dimensional for n = 2, 3, but essentially nothing is known about for n ≥ 4. For n = 4, one may construct classes by taking the cup product between two 2-classes, but it is possible that all such cup-products are trivial. We show that all such cup-products do indeed vanish if both classes are induced by the quasimorphisms of Brooks or Rolli.

Authors: Heuer, N

• DOI: 10.5287/ora-ozrjmobmm
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Bounded Cohomology; Stable Commutator Length
• Subjects: Mathematics