Hausdorff dimension of the second Grigorchuk group

Noce, Marialaura; Thillaisundaram, Anitha

doi:10.1142/s0218196721400038

“…This concept was first applied by Abercrombie [1] and by Barnea and Shalev [2] in the more general setting of profinite groups. We note that the Hausdorff dimension of the closures of several prominent weakly branch groups, such as the first [20] and second [28] Grigorchuk groups, the siblings of the first Grigorchuk group [35], the GGS-groups [15], the branch path groups [14], and generalisations of the Hanoi tower groups [33], have been computed.…”

Section: Mjommentioning

confidence: 99%

p-Basilica Groups

Domenico

¹

,

Fernández-Alcober

²

,

Noce

³

et al. 2022

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We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.

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“…The relations between a self-similar group G and the collection of portraits of its elements have already been considered in the literature, in particular by Siegenthaler in [22] and his 2008 doctoral thesis; see also [18,19,24,26]. In [20], Penland and Šunić prove that the closures of certain self-similar groups of rooted trees that satisfy an algebraic law do not have a regular language of portraits.…”

Section: Self-similar Groupsmentioning

confidence: 99%

Tree languages and branched groups

Bartholdi

¹

,

Noce

²

2023

Math. Z.

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We study the portraits of isometries of rooted trees—the labelling of the tree, at each vertex, by the permutation of its descendants—in terms of languages. We characterize regularly branched self-similar groups in terms of $$\omega $$ ω -regular languages. We deduce the algorithmic decidability of some problems, such as the comparison of regularly branched contracting groups, and their orbit structure on the boundary of the rooted tree.

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“…This concept was first applied by Abercrombie [1] and by Barnea and Shalev [2] in the more general setting of profinite groups. We note that the Hausdorff dimension of the closures of several prominent weakly branch groups, such as the first [18] and second [26] Grigorchuk groups, the siblings of the first Grigorchuk group [33], the GGS-groups [13], the branch path groups [12], and generalisations of the Hanoi tower groups [31], have been computed.…”

Section: Introductionmentioning

confidence: 99%

$p$-Basilica groups

Di Domenico,

Fernández-Alcober,

Noce

et al. 2021

Preprint

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0

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We consider a generalisation of the Basilica group to all odd primes: the p-Basilica groups acting on the p-adic tree. We show that the p-Basilica groups have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property. This provides the first examples of weakly branch groups with such properties. In addition, the p-Basilica groups give the first examples of weakly branch, but not branch, groups which are super strongly fractal. We compute the orders of the congruence quotients of these groups, which enable us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.

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Hausdorff dimension of the second Grigorchuk group

Abstract: We show that the Hausdorff dimension of the closure of the second Grigorchuk group is 43/128. Furthermore, we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normalizer in the full automorphism group of the tree.

Cited by 3 publications

References 18 publications

p-Basilica Groups

p-Basilica Groups

Tree languages and branched groups

$p$-Basilica groups

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