The space of non-extendable quasimorphisms
Morimichi Kawasaki, Mitsuaki Kimura, Shuhei Maruyama, Takahiro Matsushita, Masato Mimura

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 of a group and its normal subgroup , we consider the space of quasimorphisms and quasi-cocycles on non-extendable to . 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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Geometry and complex manifolds
