An extension theorem for quasimorphisms

TL;DR

This paper establishes a simplified and general criterion for extending quasimorphisms on subgroups, with applications to hyperbolic quotients, stable commutator lengths, and hierarchically hyperbolic groups.

Contribution

It introduces a broad sufficient condition for quasimorphism extension, simplifying previous results and providing new insights into normal subgroups and group-theoretic Dehn filling.

Findings

Extension of antisymmetric quasimorphisms when the quotient is hyperbolic

Stable commutator length is bi-Lipschitz equivalent to stable mixed commutator length

Quotients of mapping class groups by powers of pseudo-Anosov elements are hierarchically hyperbolic

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.