On the quasi-isometric classification of permutational wreath products
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915202638807040 |
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| author | Dumoncel, Vincent |
| author_facet | Dumoncel, Vincent |
| contents | In this article, we initiate the study of the large-scale geometry of permutational wreath products of the form $F\wr_{H/N}H$, where $H$ is finitely presented and where $N$ is a normal subgroup of $H$ satisfying a certain assumption of non coarse separation. The main result is a complete classification of such permutational wreath products up to quasi-isometry, building up on previous works from Genevois and Tessera. For instance, we show that, for $d\ge k\ge 2$, $\mathbb{Z}_{n}\wr_{\mathbb{Z}^{k}} \mathbb{Z}^d$ and $\mathbb{Z}_{m}\wr_{\mathbb{Z}^{k}}\mathbb{Z}^d$ are quasi-isometric if and only if $n$ and $m$ are powers of a common number. We also discuss biLipschitz equivalences between permutational wreath products, their scaling groups, as well as the quasi-isometric classification of other halo products built out of such permutational lamplighters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20159 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the quasi-isometric classification of permutational wreath products Dumoncel, Vincent Group Theory In this article, we initiate the study of the large-scale geometry of permutational wreath products of the form $F\wr_{H/N}H$, where $H$ is finitely presented and where $N$ is a normal subgroup of $H$ satisfying a certain assumption of non coarse separation. The main result is a complete classification of such permutational wreath products up to quasi-isometry, building up on previous works from Genevois and Tessera. For instance, we show that, for $d\ge k\ge 2$, $\mathbb{Z}_{n}\wr_{\mathbb{Z}^{k}} \mathbb{Z}^d$ and $\mathbb{Z}_{m}\wr_{\mathbb{Z}^{k}}\mathbb{Z}^d$ are quasi-isometric if and only if $n$ and $m$ are powers of a common number. We also discuss biLipschitz equivalences between permutational wreath products, their scaling groups, as well as the quasi-isometric classification of other halo products built out of such permutational lamplighters. |
| title | On the quasi-isometric classification of permutational wreath products |
| topic | Group Theory |
| url | https://arxiv.org/abs/2409.20159 |