Fractal dimensions and profinite groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910034399592448 |
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| author | Mayordomo, Elvira Nies, Andre |
| author_facet | Mayordomo, Elvira Nies, Andre |
| contents | Let $T$ be a finitely branching rooted tree such that any node has at least two successors. The path space $[T]$ is an ultrametric space: for distinct paths $f,g$ let $d(f,g)= 1/|T_n|$, where $T_n$ denotes the $n$-th level of the tree, and $n$ is largest such that $f(n)= g(n)$. Let $S$ be a subtree of $T$ without leaves that is level-wise uniformly branching, in the sense that the number of successors of a node only depends on its level. We~show that the Hausdorff and lower box dimensions coincide for~$[S]$, and the packing and upper box dimensions also coincide. We give geometric proofs, as well as proofs based on the point-to-set principles. We use the first result to reprove a theorem of Barnea and Shalev on the Hausdorff dimension of closed subgroups of a profinite group $G$, referring only on the geometric structure of the closed subgroup in the canonical path space given by an inverse system for $G$. We obtain an analogous theorem for the packing dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractal dimensions and profinite groups Mayordomo, Elvira Nies, Andre Group Theory Logic Metric Geometry Let $T$ be a finitely branching rooted tree such that any node has at least two successors. The path space $[T]$ is an ultrametric space: for distinct paths $f,g$ let $d(f,g)= 1/|T_n|$, where $T_n$ denotes the $n$-th level of the tree, and $n$ is largest such that $f(n)= g(n)$. Let $S$ be a subtree of $T$ without leaves that is level-wise uniformly branching, in the sense that the number of successors of a node only depends on its level. We~show that the Hausdorff and lower box dimensions coincide for~$[S]$, and the packing and upper box dimensions also coincide. We give geometric proofs, as well as proofs based on the point-to-set principles. We use the first result to reprove a theorem of Barnea and Shalev on the Hausdorff dimension of closed subgroups of a profinite group $G$, referring only on the geometric structure of the closed subgroup in the canonical path space given by an inverse system for $G$. We obtain an analogous theorem for the packing dimension. |
| title | Fractal dimensions and profinite groups |
| topic | Group Theory Logic Metric Geometry |
| url | https://arxiv.org/abs/2502.09995 |