On quasimorphisms and distortion in homeomorphism groups

Let $M$ be a smooth compact oriented connected manifold, and ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_0(M,\mu )$ the group of homeomorphisms of $M$ supported away from $\mathrm{\partial }M,$ 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 $\mathrm{\Psi }\text{\hspace{0.17em}⁣}:{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0(M,\mu )\to R$ which extend $C^0$-continuously to ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_0(M,\mu )$ as quasimorphisms, and to ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_0(M)$ 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 ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_0(M,\mu )$, and conditions under which a homeomorphism in ${\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_0(M)$ is undistorted.