On the Basilica Operation
| Autor: | Petschick, Jan Moritz, Rajeev, Karthika |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | Inspired by the Basilica group $\mathcal B$, we describe a general construction which allows us to associate to any group of automorphisms $G \leq \operatorname{Aut}(T)$ of a rooted tree $T$ a family of Basilica groups $\operatorname{Bas}_s(G), s \in \mathbb{N}_+$. For the dyadic odometer $\mathcal{O}_2$, one has $\mathcal B = \operatorname{Bas}_2(\mathcal{O}_2)$. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling $\operatorname{Bas}_s(G)$, in case $G$ fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain $\mathsf{GGS}$-groups and of generalisations of the odometer, $\mathcal{O}_m^d$. Furthermore, we study the structure of groups of type $\operatorname{Bas}_s(\mathcal{O}_m^d)$ and prove an analogue of the congruence subgroup property in the case $m = p$, a prime. Comment: 47 pages, 5 figures, to appear in to Groups Geom. Dyn. Corrections to Lemma 4.11 (ii) and subsequent statements: The assumption of being "super strongly fractal" is replaced by "very strongly fractal" |
| Databáze: | arXiv |
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