An extension theorem for quasimorphisms

Abstract

We provide a general sufficient condition for extendability of quasimorphisms on subgroups. This condition recovers the result of Hull--Osin on quasimorphisms on hyperbolically embedded subgroups, and the proof given in this paper is much simpler. We also obtain new results for quasimorphisms on normal subgroups. One result is that for a group $G$ and its normal subgroup $K$, if the quotient $G/K$ is hyperbolic, then any antisymmetric quasi-invariant quasimorphism on $K$ extends to $G$. As an application, the stable commutator length $\mathrm{scl}_G$ is bi-Lipschitz equivalent to the stable mixed commutator length $\mathrm{scl}_{G,K}$ on $[G,K]$. Another result concerns about group-theoretic Dehn filling in the sense of Dahmani--Guirardel--Osin. As an application, the quotient of a mapping class group of a surface with boundary by the normal closure of a large power of a pseudo-Anosov element is hierarchically hyperbolic. This gives an affirmative answer to a question of Fournier-Facio--Mangioni--Sisto.