On quasimorphisms and distortion in homeomorphism groups

Abstract: Let $M$ be a smooth compact oriented manifold, and $Homeo_0(M,μ)$ the group of homeomorphisms of $M$ supported away from $\partial M$, which preserve a Borel probability measure $μ$ induced by a volume form on $M$, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms $Ψ\colon Diff_0(M,μ)\to R$ which extend $C^0$-continuously to $Homeo_0(M,μ)$ as quasimorphisms, and to $Homeo_0(M)$ as group cochains whose differentials are semi-bounded cocycles. We present several applications of this result which include unboundedness of certain bi-invariant metric on the commutator subgroup of $Homeo_0(M,μ)$, and conditions under which a homeomorphism in $Homeo_0(M)$ is undistorted.