Dimensions of Zassenhaus filtration subquotients of some pro-p-groups

Mináč, Ján; Rogelstad, Michael; Tân, Nguyêñ Duy

doi:10.1007/s11856-016-1310-0

Cited by 11 publications

(17 citation statements)

References 32 publications

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“…The following corollary stands behind our definition of the sets J(n). In the case i = n it was proved by Mináč, Rogelstad and Tân [26,Cor. 3.7].…”

Section: Efratmentioning

confidence: 84%

“…I, §3] uses a specific Hall set H to give F p -linear bases of S (n,p) /S (n+1,p) for n = 2,3, as well as generating sets for arbitrary n. Namely, for similarly defined basic commutators c w ∈ S (i) of words w ∈ H with |w| = i, the generating set consists of all c p j w with n = ip j . Furthermore, according to [26,Cor. 3.12] the set of all such powers forms a basis of S (n,p) /S (n+1,p) , but the proof lacks details.…”

Section: Proposition 44mentioning

confidence: 99%

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THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

Efrat

2021

J. Inst. Math. Jussieu

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For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

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“…The following corollary stands behind our definition of the sets J(n). In the case i = n it was proved by Mináč, Rogelstad and Tân [26,Cor. 3.7].…”

Section: Efratmentioning

confidence: 84%

Section: Proposition 44mentioning

confidence: 99%

THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

Efrat

2021

J. Inst. Math. Jussieu

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“…We continue to consider the non‐soluble Demushkin pro‐

p

group

D

d

generators, with presentation , and the associated restricted

F_{p}

‐Lie algebra

D = R (D)

. In the following, we employ considerations that can be found in a very similar form, for instance, in , [, § 3] or [, § 2]. But we need more careful estimates, going somewhat further than computing the entropy of relevant Lie algebras.…”

Section: Non‐soluble Demushkin Pro‐p Groups and The Zassenhaus Seriesmentioning

confidence: 99%

“…But we need more careful estimates, going somewhat further than computing the entropy of relevant Lie algebras. We adopt, with small changes, the notation employed in . Lemma In the set‐up described above, we have

{prefixdim}_{F_{p}} false(D_{n} false) = (1 + o false(1 false)) \frac{{(d - ε)}^{n}}{n} and {prefixdim}_{F_{p}} false(sans-serifD / D_{> n} false) = (1 + o false(1 false)) \frac{{(d - ε)}^{n + 1}}{n false(d - ε - 1 false)} as n \to \infty,

where

ε \in R

with

0 < ε < 1

satisfies

(d - ε) ε = 1

.…”

Section: Non‐soluble Demushkin Pro‐p Groups and The Zassenhaus Seriesmentioning

confidence: 99%

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Pro‐p groups of positive rank gradient and Hausdorff dimension

Ocaña

Garrido

Klopsch

2019

J. London Math. Soc.

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Using Hausdorff dimension, we show that finitely generated closed subgroups H of infinite index in a finitely generated pro-p group G of positive rank gradient never contain any infinite subgroups K that are subnormal in G via finitely generated successive quotients. This generalises similar assertions that were known to hold for non-abelian free pro-p groups and other related pro-p groups.Our main results are as follows. We show that every finitely generated pro-p group G of positive rank gradient has full Hausdorff spectrum hspec F (G) = [0, 1] with respect to the Frattini series F (or, more generally, any iterated verbal filtration). Using Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-p groups and non-soluble Demushkin groups G have full Hausdorff spectrum hspec Z (G) = [0, 1] with respect to the Zassenhaus series Z. This resolves a long-standing problem in the subject.In fact, the results hold more generally for finite direct products of finitely generated prop groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-p groups and non-soluble Demushkin groups.Finally, we determine the normal Hausdorff spectra of such direct products.

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