The congruence subgroup problem for a family of branch groups

Skipper, Rachel

doi:10.1142/s0218196720500046

“…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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“…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

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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A branch group in a class of non-contracting weakly regular branch groups

Saha,

Krishna

2024

Int. J. Algebra Comput.

0

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We provide a class of non-contracting groups containing an infinite family of fractal and weakly regular branch groups, and study certain properties including abelianization, just infiniteness, and word problem. We present an example of a branch group in this class and show that it is of exponential growth. It seems this is the first example of a non-contracting branch group constructed explicitly.

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The congruence subgroup problem for a family of branch groups

Cited by 3 publications

References 16 publications

p-Basilica Groups

p-Basilica Groups

$p$-Basilica groups

A branch group in a class of non-contracting weakly regular branch groups

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