On quasimorphisms and distortion in homeomorphism groups

Abstract

Let $M$ be a smooth compact oriented connected manifold, and ${\mathrm{Homeo}}_0(M,\hat{I}¼)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ which preserve a Borel probability measure $\hat{I}¼$ induced by a volume form on $M$, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms $\hat{I}¨:{\mathrm{Diff}}_0(M,\hat{I}¼)\to R$ which extend $C^0$-continuously to ${\mathrm{Homeo}}_0(M,\hat{I}¼)$ as quasimorphisms, and to ${\mathrm{Homeo}}_0(M)$ 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 ${\mathrm{Homeo}}_0(M,\hat{I}¼)$, and conditions under which a homeomorphism in ${\mathrm{Homeo}}_0(M)$ is undistorted.

18 pages, 4 figures, accepted for publication in Comm. in Contemp. Math