The profinite completion of multi-EGS groups

Thillaisundaram, Anitha; Uria-Albizuri, Jone

doi:10.1515/jgth-2019-0155

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…We recall that a group G ≤ Aut(T ) is said to be saturated if for any n ∈ N there exists a subgroup H n ≤ St G (n) that is characteristic in G and level-transitive on every nth level subtree. Examples of saturated groups acting on rooted trees are, among others, the first Grigorchuk group [15], the p-Basilica groups [7], and the branch multi-EGS groups [23].…”

Section: Some Properties Of the Second Grigorchuk Groupmentioning

confidence: 99%

Hausdorff dimension of the second Grigorchuk group

Noce

Thillaisundaram

2021

Int. J. Algebra Comput.

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Section: Some Properties Of the Second Grigorchuk Groupmentioning

confidence: 99%

Hausdorff dimension of the second Grigorchuk group

Noce

Thillaisundaram

2021

Int. J. Algebra Comput.

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“…This gives many examples of π-free groups, among which are many branch groups [BSZ12], and in particular all Grigorchuk-Guptda-Sidki groups with non-constant defining vector [Per07,FAGUA17]. See [GUA19,TUA20] for more examples.…”

Section: π-Free Groupsmentioning

confidence: 99%

Ultrametric analogues of Ulam stability of groups

Fournier-Facio¹

2021

Preprint

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We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a p-adic analogue of Ulam stability, where we take GL n (Z p ) as approximating groups instead of U(n). For finitely presented groups, the ultrametric nature implies equivalence of the pointwise and uniform stability problems, and the profinite one implies that the corresponding approximation property is equivalent to residual finiteness. Moreover, a group is uniformly stable if and only if its largest residually finite quotient is. We provide several examples of uniformly stable groups: these include finite groups, virtually free groups, some groups acting on rooted trees, and certain lamplighter and (Generalized) Baumslag-Solitar groups.

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“…The above result gives the first examples of weakly branch, but not branch, groups that are super strongly fractal; cf. [32,Prop. 3.11] and [34,Prop.…”

Section: Commutator Subgroup Structurementioning

confidence: 99%

“…Examples of other groups acting on rooted trees with automorphism group equal to its normaliser in Aut(T ) are the Grigorchuk group and the Brunner-Sidki-Vieira group [23], and the branch multi-EGS groups [32].…”

Section: Commutator Subgroup Structurementioning

confidence: 99%

$p$-Basilica groups

Di Domenico,

Fernández-Alcober,

Noce

et al. 2021

Preprint

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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 profinite completion of multi-EGS groups

Cited by 5 publications

References 12 publications

Hausdorff dimension of the second Grigorchuk group

Hausdorff dimension of the second Grigorchuk group

Ultrametric analogues of Ulam stability of groups

$p$-Basilica groups

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