The space of non-extendable quasimorphisms

Abstract

In the present paper, 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 commutator subgroup of a Gromov hyperbolic group, the kernel of the (volume) flux homomorphism, and the IA-automorphism group of a free group. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the mixed stable commutator length for certain pairs of a group and its normal subgroup.