On quasimorphisms and distortion in homeomorphism groups
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909013945352192 |
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| author | Brandenbursky, Michael Kedra, Jarek Marcinkowski, Michal Shelukhin, Egor |
| author_facet | Brandenbursky, Michael Kedra, Jarek Marcinkowski, Michal Shelukhin, Egor |
| contents | Let $M$ be a smooth compact oriented connected manifold, and ${\rm Homeo}_0(M,μ)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ 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 $Ψ\colon {\rm Diff}_0(M,μ)\to R$ which extend $C^0$-continuously to ${\rm Homeo}_0(M,μ)$ as quasimorphisms, and to ${\rm Homeo}_0(M)$ 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 ${\rm Homeo}_0(M,μ)$, and conditions under which a homeomorphism in ${\rm Homeo}_0(M)$ is undistorted. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On quasimorphisms and distortion in homeomorphism groups Brandenbursky, Michael Kedra, Jarek Marcinkowski, Michal Shelukhin, Egor Geometric Topology Dynamical Systems Group Theory Let $M$ be a smooth compact oriented connected manifold, and ${\rm Homeo}_0(M,μ)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ 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 $Ψ\colon {\rm Diff}_0(M,μ)\to R$ which extend $C^0$-continuously to ${\rm Homeo}_0(M,μ)$ as quasimorphisms, and to ${\rm Homeo}_0(M)$ 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 ${\rm Homeo}_0(M,μ)$, and conditions under which a homeomorphism in ${\rm Homeo}_0(M)$ is undistorted. |
| title | On quasimorphisms and distortion in homeomorphism groups |
| topic | Geometric Topology Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2512.15375 |