The space of non-extendable quasimorphisms

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.Comment: 58 pages, 1 figure. Major revision. Theorem 1.12 in v3 has been generalized to Theorem 1.2 in the current version: this new theorem treats hyperbolic mapping tori in general cases, and it serves as a leading application of our main theore