Normal Subgroups of Profinite Groups of Finite Cohomological Dimension

Engler, Antonio José; Haran, Dan; Kochloukova, Dessislava H.; Zalesskiĭ, Pavel

doi:10.1112/s0024610703005003

Cited by 13 publications

(8 citation statements)

References 11 publications

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“…It is well known that a surface group is p-good for all primes p (cf. (iii) of Lemma 5.12 in [9], for instance). This implies that lim − →H/K∈S H k (K, Z/p) = 0, for k > 0 and all primes p ∈ Λ S .…”

Section: A Geometric Proof Of Theorem 15mentioning

confidence: 94%

A restricted Magnus property for profinite surface groups

Boggi

Zalesskiĭ²

2018

Trans. Amer. Math. Soc.

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Magnus [16] proved that, given two elements x and y of a finitely generated free group F with equal normal closures x F = y F , then x is conjugated either to y or y −1 . More recently ([5] and [6]), this property, called the Magnus property, has been generalized to oriented surface groups.In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for S a class of finite groups, we prove that, if x and y are algebraically simple (cf. Definition 1.3) elements of the pro-S completion Π S of an orientable surface group Π, such that, for all n ∈ N, there holds x n Π S = y n Π S , then x is conjugated to y s for some s ∈ ( Z S ) * . As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions.The most important application of the theory above is a generalisation of the description of centralizers of profinite Dehn twists given in [3] to profinite Dehn multitwists.

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Section: A Geometric Proof Of Theorem 15mentioning

confidence: 94%

A restricted Magnus property for profinite surface groups

Boggi

Zalesskiĭ²

2018

Trans. Amer. Math. Soc.

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“…The only place its proof relies on Theorem 6.5 in [5] is the fact that in Theorem 4.1 N cannot be procyclic. But if N was procyclic then by [4] by substituting H with a subgroup of finite index we can assume that H/N has finite cohomological dimension, thus χ(H/N ) is well defined. Then χ(H) = χ(N )χ(H/N ) = 0 and we are done as in the previous proof.…”

Section: Subgroups and Homology Of Extensions Of Centralizers Of Pro-...mentioning

confidence: 99%

Subgroups and homology of extensions of centralizers of pro-pgroups

Kochloukova

Zalesskiĭ

2014

Math. Nachr.

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“…If K is analytic pro-p of dimension d ≥ 1, then G/K again has finite vcd, by [9, Theorem 0.7]. There are more such results in [9].…”

Section: Iterated Homotopy Fixed Points For Profinite Galois Extensionsmentioning

confidence: 99%

Function spectra and continuous G -spectra

Davis¹

2011

Bulletin of the London Mathematical Society

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Let G be a profinite group, {Xα}α be a cofiltered diagram of discrete G‐spectra, and Z be a spectrum with trivial G‐action. We show how to define the homotopy fixed point spectrum F(Z, holimα Xα)hG and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holimα (Xα)hG). With these tools, we show that the K(n)‐local Spanier–Whitehead dual is always a homotopy fixed point spectrum, a well‐known Adams‐type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k‐local profinite G‐Galois extension E, with K ◃ G and closed, the equivalence (EhkK)hkG/K ≃ EhkG (due to Behrens and the author), where (−)hk(−) denotes k‐local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non‐local) homotopy fixed points, when G/K has finite vcd.

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Normal Subgroups of Profinite Groups of Finite Cohomological Dimension

Cited by 13 publications

References 11 publications

A restricted Magnus property for profinite surface groups

A restricted Magnus property for profinite surface groups

Subgroups and homology of extensions of centralizers of pro-pgroups

Function spectra and continuous G -spectra

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