Pro-
                
                  
                
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               congruence properties for groups of rooted tree automorphisms

Garrido, Alejandra; Uria-Albizuri, Jone

doi:10.1007/s00013-018-1278-6

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…The class of all finite p-groups is a well-behaved class, i.e., it is closed under taking subgroups, quotients, extensions and direct limits. In light of this, we prove that Bas s .O d p / has the p-congruence subgroup property (p-CSP), a weaker version of CSP introduced by Garrido and Uria-Albizuri in [18]. The group G has the p-CSP if every subgroup of index a power of p in G contains some layer stabiliser in G. In [18] one finds a sufficient condition for a weakly branch group to have the p-CSP and it is also proved that the original Basilica group B has the 2-CSP.…”

Section: Introductionmentioning

confidence: 89%

“…In light of this, we prove that Bas s .O d p / has the p-congruence subgroup property (p-CSP), a weaker version of CSP introduced by Garrido and Uria-Albizuri in [18]. The group G has the p-CSP if every subgroup of index a power of p in G contains some layer stabiliser in G. In [18] one finds a sufficient condition for a weakly branch group to have the p-CSP and it is also proved that the original Basilica group B has the 2-CSP. This argument is general-ised by Di Domenico, Fernández-Alcober, Noce and Thillaisundaram [13] to see that the p-Basilica groups have the p-CSP.…”

Section: Introductionmentioning

confidence: 89%

“…Here we prove that the generalised Basilica group Bas s .O d p / has the p-CSP for d; s 2 N C with s > 2 and p a prime. We follow the strategy from [18], where it is proved that the original Basilica group B D Bas 2 .O 2 / has the 2-congruence subgroup property. However, on account of Theorem 7.2 and Lemma 7.3, our reasoning must be different, and we will use Theorem 5.1.…”

Section: Congruence Properties Of the Generalised Basilica Groupsmentioning

confidence: 99%

“…It was the first group known to be not sub-exponentially amenable [21], but amenable [7,11]. Furthermore, it is the iterated monodromy group of z 2 1 [22,26], and it has the 2-congruence subgroup property [18].…”

Section: Introductionmentioning

confidence: 99%

“…Even though we follow the same strategy as in [18], the arguments differ significantly because of Theorem 1.9. Here we make use of Theorem 1.4 to obtain a normal generating set for the layer stabilisers of the generalised Basilica groups (Theorem 5.1).…”

Section: Introductionmentioning

confidence: 99%

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On the Basilica operation

Moritz

Rajeev

2023

Groups Geom. Dyn.

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Inspired by the Basilica group B, we describe a general construction which allows us to associate to any group of automorphisms G Ä Aut.T / of a rooted tree T a family of Basilica groups Bas s .G/, s 2 N C . For the dyadic odometer O 2 , one has B D Bas 2 .O 2 /. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling 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 GGS-groups and of generalisations of the odometer, O d m . Furthermore, we study the structure of groups of type Bas s .O d m / and prove an analogue of the congruence subgroup property in the case m D p, a prime.

show abstract

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 89%

Section: Congruence Properties Of the Generalised Basilica Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Basilica operation

Moritz

Rajeev

2023

Groups Geom. Dyn.

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p-Basilica Groups

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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On maximal subgroups of infinite index in branch and weakly branch groups

Francoeur¹

2020

Journal of Algebra

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Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

Cited by 5 publications

References 12 publications

On the Basilica operation

On the Basilica operation

p-Basilica Groups

On maximal subgroups of infinite index in branch and weakly branch groups

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