Subgroups of prescribed finite index in linear groups

Wehrfritz, B. A. F.

doi:10.1007/bf02764674

Cited by 3 publications

(3 citation statements)

References 3 publications

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“…Proof. The first item is a special case of a more general result in [37]. For the latter, it is clearly enough to prove that every infinite residually finite 2-group has a subgroup of even finite index.…”

Section: Lemma 39 the Following Groups Have Halvable Finite-index Sub...mentioning

confidence: 94%

Gate lattices and the stabilized automorphism group

Salo

2023

JMD

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We study the stabilized automorphism group of a subshift of finite type with a certain gluing property called the eventual filling property, on a residually finite group G. We show that the stabilized automorphism group is simply monolithic, i.e., it has a unique minimal non-trivial normal subgroupthe monolith-which is additionally simple. To describe the monolith, we introduce gate lattices, which apply (reversible logical) gates on finite-index subgroups of G. The monolith is then precisely the commutator subgroup of the group generated by gate lattices. If the subshift and the group G have some additional properties, then the gate lattices generate a perfect group, thus they generate the monolith. In particular, this is always the case when the acting group is the integers. In this case we can also show that gate lattices generate the inert part of the stabilized automorphism group. Thus we obtain that the stabilized inert automorphism group of a one-dimensional mixing subshift of finite type is simple.

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Section: Lemma 39 the Following Groups Have Halvable Finite-index Sub...mentioning

confidence: 94%

Gate lattices and the stabilized automorphism group

Salo

2023

JMD

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“…Proof. The first item is a special case of a more general result of [15]. For the latter, it is clearly enough to prove that every infinite residually finite 2-group has a subgroup of even finite index.…”

Section: Lemma 19 the Following Groups Have Halvable Subgroupsmentioning

confidence: 95%

“…In particular, full shifts have a simple L if their alphabet is even, or the group has halvable subgroups. All finitely-generated infinite finite-dimensional matrix groups over commutative rings have halvable subgroups by [15], and so do (trivially) all infinite residually finite 2-groups, including the (non-linear) Grigorchuk group [6].…”

Section: Main Statementsmentioning

confidence: 99%

Gate lattices

Salo¹

2022

Preprint

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A reversible gate on a subshift on a residually finite group G can be applied on any sparse enough finite-index subgroup H, to obtain what we call a gate lattice. Gate lattices are automorphisms of the shift action of H, thus generate a subgroup of the Hartman-Kra-Schmieding stabilized automorphism group. We show that for subshifts of finite type with a gluing property we call the eventual filling property, the subgroup generated by even gate lattices is simple. Under some conditions, even gate lattices generate all gate lattices, and in the case of a one-dimensional mixing SFT, they generate the inert part of the stabilized automorphism group, thus we obtain that this group is simple. In the case of a full shift this has been previously shown by Hartman, Kra and Schmieding.

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Growth in Infinite Groups of Infinite Subsets

Button

2015

Algebra Colloq.

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Given an infinite group G, we consider the finitely additive measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on Ruzsa distance and product free sets. In particular if G has infinitely many finite index subgroups then it has subsets S of measure arbitrarily close to 1/2 with square S 2 having measure less than 1.

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Subgroups of prescribed finite index in linear groups

Cited by 3 publications

References 3 publications

Gate lattices and the stabilized automorphism group

Gate lattices and the stabilized automorphism group

Gate lattices

Growth in Infinite Groups of Infinite Subsets

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