On Quasimorphisms and Distortion in Homeomorphism Groups

Let [Formula: see text] be a smooth compact oriented connected manifold, and [Formula: see text] the group of homeomorphisms of [Formula: see text] supported away from [Formula: see text] which preserve a Borel probability measure [Formula: see text] induced by a volume form on [Formula: see text], and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms [Formula: see text] which extend [Formula: see text]-continuously to [Formula: see text] as quasimorphisms, and to [Formula: see text] 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 [Formula: see text], and conditions under which a homeomorphism in [Formula: see text] is undistorted.