Isoperimetric profiles of lamplighter-like groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908968593391616 |
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| author | Correia, Corentin Dumoncel, Vincent |
| author_facet | Correia, Corentin Dumoncel, Vincent |
| contents | Given a finitely generated amenable group $H$ satisfying some mild assumptions, we relate isoperimetric profiles of the lampshuffler group $\mathsf{Shuffler}(H)=\mathsf{FSym}(H)\rtimes H$ to those of $H$. Our results are sharp for all exponential growth groups for which isoperimetric profiles are known, including Brieussel-Zheng groups. This refines previous estimates obtained by Erschler and Zheng and by Saloff-Coste and Zheng. The most difficult part is to find an optimal upper bound, and our strategy consists in finding suitable lamplighter subgraphs in lampshufflers. This novelty applies more generally for many examples of halo products, a class of groups introduced recently by Genevois and Tessera as a natural generalisation of wreath products. Lastly, we also give applications of our estimates on isoperimetric profiles to the existence problem of regular maps between such groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isoperimetric profiles of lamplighter-like groups Correia, Corentin Dumoncel, Vincent Group Theory Geometric Topology Metric Geometry 20F65, 20F69 Given a finitely generated amenable group $H$ satisfying some mild assumptions, we relate isoperimetric profiles of the lampshuffler group $\mathsf{Shuffler}(H)=\mathsf{FSym}(H)\rtimes H$ to those of $H$. Our results are sharp for all exponential growth groups for which isoperimetric profiles are known, including Brieussel-Zheng groups. This refines previous estimates obtained by Erschler and Zheng and by Saloff-Coste and Zheng. The most difficult part is to find an optimal upper bound, and our strategy consists in finding suitable lamplighter subgraphs in lampshufflers. This novelty applies more generally for many examples of halo products, a class of groups introduced recently by Genevois and Tessera as a natural generalisation of wreath products. Lastly, we also give applications of our estimates on isoperimetric profiles to the existence problem of regular maps between such groups. |
| title | Isoperimetric profiles of lamplighter-like groups |
| topic | Group Theory Geometric Topology Metric Geometry 20F65, 20F69 |
| url | https://arxiv.org/abs/2506.13235 |