Maximal subgroups of multi-edge spinal groups

Alexoudas, Theofanis; Klopsch, Benjamin; Thillaisundaram, Anitha

doi:10.4171/ggd/359

Cited by 28 publications

(100 citation statements)

References 11 publications

(35 reference statements)

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“…Since the commutator subgroup is verbal, we see that Finally, consider the general virtually soluble case. The argument used to prove (1) shows that there is a soluble subgroup D of finite index in B such that φ(D) ≤ D. We know that D ∞ ∈ M and h(D ∞ ) ≤ h(D) by the soluble case, and the result follows since D ∞ has finite index in B ∞ .…”

Section: Applications To Generalized Baumslag-solitar Groupsmentioning

confidence: 96%

On the residual and profinite closures of commensurated subgroups

Caprace¹,

Kropholler²,

Reid³

et al. 2019

Math. Proc. Camb. Phil. Soc.

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The residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. Various applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups are described.

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Section: Applications To Generalized Baumslag-solitar Groupsmentioning

confidence: 96%

On the residual and profinite closures of commensurated subgroups

Caprace¹,

Kropholler²,

Reid³

et al. 2019

Math. Proc. Camb. Phil. Soc.

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“…, b r . We refer the reader to the paper [2] (i) Every infinitely iterated wreath product of finite transitive permutation groups of degree at least 2. In particular, Aut T and Γ p , for p a prime.…”

Section: Left Engel Elements In Weakly Branch Groupsmentioning

confidence: 99%

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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“…. , p} with r j = 0, the subgroup G j = a, b (j) ≤ G is a multi-edge spinal group, and [1,Proposition 3.5] shows that…”

Section: Proof By Spherical Transitivity It Suffices To Show Thatmentioning

confidence: 99%

“…Next we record the following result regarding the derived subgroup G ′ of G, based on the extra assumption that there are sufficiently many nonsymmetric defining vectors. (1) , . .…”

Section: Proof By Spherical Transitivity It Suffices To Show Thatmentioning

confidence: 99%

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Maximal Subgroups and Irreducible Representations of Generalized Multi-Edge Spinal Groups

Thillaisundaram

Klopsch

2018

Proceedings of the Edinburgh Mathematical Society

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Let p ≥ 3 be a prime. A generalised multi-edge spinal groupis a subgroup of the automorphism group of a regular p-adic rooted tree T that is generated by one rooted automorphism a and p families br j of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families.This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multiedge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore we use tree enveloping algebras, which were introduced by Sidki and Bartholdi, to show that certain generalised multi-edge spinal groups admit faithful infinite dimensional irreducible representations over the prime field Z/pZ.

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Maximal subgroups of multi-edge spinal groups

Cited by 28 publications

References 11 publications

On the residual and profinite closures of commensurated subgroups

On the residual and profinite closures of commensurated subgroups

Engel elements in weakly branch groups

Maximal Subgroups and Irreducible Representations of Generalized Multi-Edge Spinal Groups

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