Cup Product in Bounded Cohomology of the Free Group

TL;DR

This paper investigates the bounded cohomology of free groups, focusing on the cup product in degree 4, and shows that certain classes constructed from degree 2 classes are trivial.

Contribution

It demonstrates that cup products of specific degree 2 classes in bounded cohomology of free groups are trivial, advancing understanding of the structure of bounded cohomology.

Findings

Cup products of Brooks or Rolli classes in degree 2 are trivial in degree 4.

Full bounded cohomology of free groups remains largely unknown for degrees ≥ 4.

The paper provides new insights into the algebraic structure of bounded cohomology.

Abstract

The theory of bounded cohomology of groups has many applications. A key open problem is to compute the full bounded cohomology $H_b^n(F, R)$ of a non-abelian free group $F$ with trivial real coefficients. It is known that $H_b^n(F,R)$ is trivial for $n=1$ and uncountable dimensional for $n=2,3$, but remains unknown for any $n \geq 4$. For $n=4$, one may construct classes by taking the cup product $\alpha \cup \beta \in H_b^4(F, R)$ between two $2$-classes $\alpha, \beta \in H^2_b(F, R)$. However, we show that all such cup products are trivial if $\alpha$ and $\beta$ are classes induced by the quasimorphisms defined by Brooks or Rolli.