On the critical regularity of nilpotent groups acting on the interval: the metabelian case
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915190175432704 |
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| author | Escayola, Maximiliano Rivas, Cristóbal |
| author_facet | Escayola, Maximiliano Rivas, Cristóbal |
| contents | Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian group. In this work we show that $G$ embeds into the group of orientation preserving $C^{1+α}$-diffeomorphisms of the compact interval, for all $α< 1/k$ where $k$ is the torsion-free rank of $G/A$ and $A$ is a maximal abelian subgroup. We show that in many situations the corresponding $1/k$ is critical in the sense that there is no embedding of $G$ with higher regularity. A particularly nice family where this happens, is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity equals $1+1/n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00342 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the critical regularity of nilpotent groups acting on the interval: the metabelian case Escayola, Maximiliano Rivas, Cristóbal Group Theory Dynamical Systems 20F18, 37C05, 37C85, 37E05 Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian group. In this work we show that $G$ embeds into the group of orientation preserving $C^{1+α}$-diffeomorphisms of the compact interval, for all $α< 1/k$ where $k$ is the torsion-free rank of $G/A$ and $A$ is a maximal abelian subgroup. We show that in many situations the corresponding $1/k$ is critical in the sense that there is no embedding of $G$ with higher regularity. A particularly nice family where this happens, is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity equals $1+1/n$. |
| title | On the critical regularity of nilpotent groups acting on the interval: the metabelian case |
| topic | Group Theory Dynamical Systems 20F18, 37C05, 37C85, 37E05 |
| url | https://arxiv.org/abs/2305.00342 |