The space of non-extendable quasimorphisms

TL;DR

This paper investigates the space of non-extendable quasimorphisms on a normal subgroup within a group, establishing a cohomological framework and applying it to various algebraic structures to relate different notions of commutator length.

Contribution

It introduces a five-term exact sequence in cohomology for non-extendable quasimorphisms and applies it to several important group-theoretic contexts.

Findings

Established a cohomological exact sequence for non-extendable quasimorphisms.

Connected stable commutator length with stable mixed commutator length in specific groups.

Applied the theory to flux homomorphisms, IA-automorphisms, and hyperbolic groups.

Abstract

For a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the space of quasimorphisms and quasi-cocycles on $N$ non-extendable to $G$. To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As its application, we study the spaces associated with the kernel of the (volume) flux homomorphism, the IA-automorphism group of a free group, and certain normal subgroups of Gromov-hyperbolic groups. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the stable mixed commutator length for certain pairs of a group and its normal subgroup.