On quasimorphisms and distortion in homeomorphism groups

Let $M$ be a smooth compact oriented connected manifold, and ${\rm Homeo}_0(M,\mu)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ which preserve a Borel probability measure $\mu$ induced by a volume form on $M$, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms $\Psi\colon {\rm Diff}_0(M,\mu)\to R$ which extend $C^0$-continuously to ${\rm Homeo}_0(M,\mu)$ as quasimorphisms, and to ${\rm Homeo}_0(M)$ 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 ${\rm Homeo}_0(M,\mu)$, and conditions under which a homeomorphism in ${\rm Homeo}_0(M)$ is undistorted.