Second countable virtually free pro-p groups whose torsion elements have finite centralizer

MacQuarrie, John; Zalesskiĭ, Pavel

doi:10.1007/s00029-016-0230-5

Cited by 4 publications

(9 citation statements)

References 15 publications

(32 reference statements)

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“…Our goal for this section is to prove Theorem 1.2. A special case of this result was proved in [23] with R = Z p . The missing link needed to obtain the result in generality is Theorem 8.6, a generalization of [36,Thm 3] for infinitely generated lattices.…”

Section: Weiss' Theoremmentioning

confidence: 82%

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

MacQuarrie

Symonds

Zalesskiĭ

2020

Advances in Mathematics

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2 Definitions, terminology and background 2.1 Algebras and modules Our coefficient ring R will always be a commutative pseudocompact ring. In later sections we will require further structure on R, the main coefficient rings of interest to us being complete discrete valuation rings.Let Λ be a pseudocompact R-algebra (we follow the treatments in [9] and [18]). Examples of particular interest are the completed group algebra R[[G]] of a profinite group G, or later the group algebra RG of a finite group G. We consider the following categories of modules for Λ:• Λ-Mod C : the category whose objects are pseudocompact left Λ-modules [9, §1]. Objects of this category are complete, Hausdorff topological Λ-modules U having a basis of open neighbourhoods of 0 consisting of submodules V for which U/V has finite length. In other words, the category of inverse limits of left Λ-modules of finite length over R.• Λ-Mod D : the category whose objects are those topological Λ-modules having the discrete topology. Such modules are precisely the direct limits of left Λ-modules of finite length over R. We will call such modules discrete (they are called "locally finite" in [18]).• Λ-Mod abs : the category of abstract left Λ-modules Proof. We mimic the proof of [38, Lem. 52.2]. Write X = I X i with each X i an RG-sublattice of X of R-rank 1. The sequence remains split over R and the X i are indecomposable R-modules, so by Proposition 3.5 applied to these R-modules there is a subset J of I such that X = s(X ′ ) ⊕ J X i , and this remains true over RG. Thus the RG-module homomorphism h ′ = h| J Xi is an isomorphism, where h is the map induced by h. By Nakayama's Lemma, h| J Xi = h ′ : J X i → X ′′ is an isomorphism. Now the composition of h ′−1 with the inclusion J X i → X is a splitting of h.

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Section: Weiss' Theoremmentioning

confidence: 82%

Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

MacQuarrie

Symonds

Zalesskiĭ

2020

Advances in Mathematics

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“…Observe that if

N ⩽ H

then

{double-struckZ}_{p} {[G / H]}^{N} = {double-struckZ}_{p} false[G / H false]

. Otherwise

{double-struckZ}_{p} false[G / H false]

{double-struckZ}_{p} false[N false]

‐free and so

{double-struckZ}_{p} {[G / H]}^{N} = {double-struckZ}_{p} {[G / H]}_{N} = {double-struckZ}_{p} false[G / H N false]

(see [, Lemma 2.4]).…”

Section: Preliminariesmentioning

confidence: 99%

“…We can apply Lemma and the induction hypothesis to

G / C^{G}

to deduce that

F_{C}^{a b}

is an

{double-struckZ}_{p} false[H / C false]

‐permutation module. Now

F_{C}^{a b} / M_{1} ≅ {(M_{p})}_{C}

and by [, Lemma 2.4]

{(M_{p})}_{C} ≅ M_{p}^{C}

, so

{(F^{a b})}^{C} ≅ F_{C}^{a b}

is a permutation

{double-struckZ}_{p} false[H / C false]

‐module. Thus hypothesis (ii) of Theorem is also satisfied and so

F^{a b}

is a permutation

{double-struckZ}_{p} false[H false]

‐module.

□

…”

Section: The Structure Of the Abelianizationmentioning

confidence: 99%

See 1 more Smart Citation

Infinitely generated virtually free pro‐p groups and p‐adic representations

Zalesskiĭ

2018

Journal of Topology

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We prove the pro‐p version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro‐p group G has finite centralizers of all non‐trivial torsion elements, a stronger statement is proved: G embeds into a free pro‐p product of a free pro‐p group and a finite p‐group. Integral p‐adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro‐p case.

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“…Remark The proposition is valid for second countable pro‐