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.
- Pub Date:
- July 2021
- Mathematics - Group Theory;
- Mathematics - Functional Analysis;
- Mathematics - Geometric Topology;
- Mathematics - Symplectic Geometry
- 48 pages, 1 figures. We add Subsection 8.3, where we describe a relation between the 2-Kazhdan property and the space of non-extendable quasi-cocycles