A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913653022785536 |
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| author | Radi, Santiago |
| author_facet | Radi, Santiago |
| contents | This article provides a method to calculate the fixed-point proportion of any iterated wreath product acting on a $d$-regular tree. Moreover, the method applies to a generalization of iterated wreath products acting on a $d$-regular tree, which are not groups. As an application of this generalization, a family of groups of finite type of depth $2$ acting on a $d$-regular tree with $d \geq 3$ and $d \neq 2 \pmod{4}$ is constructed. These groups are self-similar, level-transitive, have positive Hausdorff dimension, and exhibit a positive fixed-point proportion. Unlike other groups with a positive fixed-point proportion known in the literature, the fixed-point proportion of this new family can be calculated explicitly. Furthermore, the iterated Galois group of the polynomial $x^d + 1$ with $d \geq 2$ appears in this family, so its fixed-point proportion is calculated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00515 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension Radi, Santiago Group Theory Number Theory This article provides a method to calculate the fixed-point proportion of any iterated wreath product acting on a $d$-regular tree. Moreover, the method applies to a generalization of iterated wreath products acting on a $d$-regular tree, which are not groups. As an application of this generalization, a family of groups of finite type of depth $2$ acting on a $d$-regular tree with $d \geq 3$ and $d \neq 2 \pmod{4}$ is constructed. These groups are self-similar, level-transitive, have positive Hausdorff dimension, and exhibit a positive fixed-point proportion. Unlike other groups with a positive fixed-point proportion known in the literature, the fixed-point proportion of this new family can be calculated explicitly. Furthermore, the iterated Galois group of the polynomial $x^d + 1$ with $d \geq 2$ appears in this family, so its fixed-point proportion is calculated. |
| title | A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension |
| topic | Group Theory Number Theory |
| url | https://arxiv.org/abs/2501.00515 |