Higher-Degree Bounded Cohomology of Non-Abelian Free Groups

Determine whether the bounded cohomology Hb^n(F; R) of a non-abelian free group F with trivial real coefficients is trivial in all degrees n greater than 3.

Background

The paper discusses the difficulty of computing bounded cohomology of non-abelian free groups F. While it is known that Hb2(F; R) is infinite dimensional via quasimorphisms and Hb3(F; R) is infinite dimensional via hyperbolic geometry, the structure in higher degrees remains mysterious.

This long-standing question, also recorded as [6, Question 16.3], motivates attempts to understand higher-degree classes via products with degree-2 classes, which are represented by coboundaries of quasimorphisms.

References

We know that bounded cohomology of F is infinite dimensional in degree 2 via quasimorphisms [3] and 3 via hyperbolic geometry [23], but for now it is unknown whether it is trivial in higher degrees or not [6, Question 16.3].

— A vanishing criterion for cup products and Massey products in bounded cohomology  (2407.17034 - Hofmann, 2024) in Section 1. Introduction (page 4)

However, understanding the bounded cohomology of $F_2$ in degrees greater than three remains a fundamental open problem; in particular, infinite-dimensionality of its reduced bounded cohomology in these degrees is not known.

— Bounded cohomology of transformation groups in all degrees  (2610.11327 - Brandenbursky, 8 Oct 2026) in Section 1, Introduction