Engel elements in some fractal groups

Fernández-Alcober, Gustavo A.; Garreta, Albert; Noce, Marialaura

doi:10.1007/s00605-018-1218-3

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…Furthermore, the Julia set of z 2 − 1, which is the set of accumulation points of the backward iterations of an arbitrary point in the complex plane under z 2 − 1, can be approximated by a sequence of finite Schreier graphs obtained by the action of the Basilica group at each level of the binary tree; see [12], where all limits of finite Schreier graphs of the Basilica group were classified up to isomorphism. The Basilica group has also been studied in other contexts in group theory, for example in [13] it has been proved that the Basilica group has no nontrivial Engel elements.…”

Section: Mjommentioning

confidence: 99%

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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Section: Mjommentioning

confidence: 99%

p-Basilica Groups

et al. 2022

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“…. , 1) then L(G E ) = 1 by [7,Theorem 7]. Otherwise G E is a branch group, and the result follows immediately from Theorem B and from the characterisation of periodic multi-GGS groups given above.…”

Section: Left Engel Elements In Weakly Branch Groupsmentioning

confidence: 78%

“…In the aforementioned paper, Bartholdi also proved that, if G is the Gupta-Sidki 3-group, then L(G) = 1. On the other hand, in [7], Garreta and the first two authors proved that again L(G) = 1 if G is any fractal subgroup of a Sylow pro-p subgroup of the group of automorphisms of the p-adic tree satisfying the condition |G : st G (1) | = ∞, and in particular if G is non-abelian and has torsion-free abelianization. Also, Tortora and the second author showed in [16] that L(G) = R(G) = 1 for the Grigorchuk group G. As we next see, the situation in these classes of groups generalises to a great extent to weakly branch groups, which have a tendency to have trivial Engel sets.…”

Section: Introductionmentioning

confidence: 99%

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Engel elements in weakly branch groups

2020

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We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all pelements and that the group is virtually a p-group (and so periodic) for some prime p. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these results to well-known families of weakly branch groups, like the multi-GGS groups.2010 Mathematics Subject Classification. 20E08, 20F45.

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“…We refer to the surveys [1,16] for further general information. See also [7,15] for recent results on right and bounded Engel elements in some types of branch groups.…”

Section: Introductionmentioning

confidence: 99%