On quasimorphisms and distortion in homeomorphism groups

2025/12/17 by Brandenbursky, Michael, Kedra, Jarek, Marcinkowski, Michal +1
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2512.15375

Abstract

Let M be a smooth compact oriented manifold, and Homeo₀(M,μ) the group of homeomorphisms of M supported away from ∂ M, which preserve a Borel probability measure μ induced by a volume form on M, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms Ψ\colon Diff₀(M,μ)→ R which extend C⁰-continuously to Homeo₀(M,μ) as quasimorphisms, and to Homeo₀(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 Homeo₀(M,μ), and conditions under which a homeomorphism in Homeo₀(M) is undistorted.

Citations

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Related

• Large scale geometry of homeomorphism groups
• Fragmentation norm and relative quasimorphisms
• Entropy and quasimorphisms
• Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band
• Reconstructing maps out of groups