Certain Gambaudo-Ghys and Polterovich quasimorphisms extend C^0-continuously to Homeo_0(M,μ) as quasimorphisms and to Homeo_0(M) as cochains with semi-bounded differentials, implying unbounded metrics on commutator subgroups and conditions for undistorted homeomorphisms.
The $L^p$-diameter of the space of contractible loops
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abstract
We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a special case, this resolves the $L^p$-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms.
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On quasimorphisms and distortion in homeomorphism groups
Certain Gambaudo-Ghys and Polterovich quasimorphisms extend C^0-continuously to Homeo_0(M,μ) as quasimorphisms and to Homeo_0(M) as cochains with semi-bounded differentials, implying unbounded metrics on commutator subgroups and conditions for undistorted homeomorphisms.