Pro-p Groups Research Articles

The base change in the Atiyah and the Lück approximation conjectures

Let F be a free finitely generated group and $${A \in {\rm Mat}_{n \times m}(\mathbb{C}[F])}$$ . For each quotient G = F/N of F we can define a von Neumann rank function rkG(A) associated with the l2-operator l2(G)n → l2(G)m induced by right multiplication by A. For example, in the case where G is finite, $${{\rm rk}_G(A)=\frac{{\rm rk}_{\mathbb{C}}({{\bar{A}}})}{|G|}}$$ is the normalized rank of the matrix $${{{\bar{A}}} \in {\rm Mat}_{n \times m}(\mathbb{C}[G])}$$ obtained by reducing the coefficients of A modulo N. One of the variations of the Luck approximation conjecture claims that the function $${N\mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of G is finite, then $${{\rm rk}_G(A) \in \frac{1}{lcm (G)}\mathbb{Z}}$$ . In our first result we prove the sofic Luck approximation conjecture. In particular, we show that the function $${N \mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Luck approximation and we show that the strong Atiyah conjecture holds for groups from a class $${{\mathcal{D}}}$$ , virtually compact special groups, Artin’s braid groups and torsion-free p-adic analytic pro-p groups.

Cerebral Oxygenation During Neonatal Intubation-Ancillary Study of the Prettineo-Study.

Purpose: This study aimed to describe cerebral Near InfraRed Spectroscopy (NIRS) profiles during neonatal intubation using two different premedication regimens.Methods: Neonates requiring non-emergency intubation were enrolled in an ancillary study, conducted in two French Neonatal Intensive Care Units participating in a larger on-going multicenter, double blind, randomized, controlled trial. Patients were randomly assigned to the “atropine-propofol” (Prop) group or the “atropine-atracurium-sufentanil” (SufTrac) group. Regional cerebral oxygen saturation (rScO2), pulse oxymetry (SpO2), mean arterial blood pressure (MABP), and transcutaneous partial pressure of carbon dioxide (TcPCO2) were collected at 9 predefined time points from 1 min before to 60 min after the first drug injection. The two primary outcomes were a decrease in rScO2 value >20% from baseline and a decrease in fractional cerebral tissue oxygen extraction (FTOE) value >10% from baseline, at any time point. Secondary outcomes included physiological parameters changes over time and correlations between mean arterial blood pressure, and FTOE at different time points. Descriptive results were obtained and exploratory statistical analyses were performed for 24 included patients.Results: rScO2 decreased in 5/11 (46%) infants from the Prop group and 10/13 (77%) from the SufTrac group (p = 0.11); FTOE decreased in 10/11 (91%) infants from the Prop group, and 12/13 (92%) from the SufTrac group (p = 0.90). rScO2 values decreased over time in both groups, whereas FTOE's pattern appeared more stable. SpO2 and transcutaneous TcPCO2 seemed more preserved in the Prop group while MABP seemed more preserved in the SufTrac group. No important correlation was observed between MABP and FTOE (r = 0.08 to 0.12 across the time points).Conclusion: Our results suggest a frequent decrease in cerebral oxygenation without obvious impairment in cerebral autoregulation during neonatal intubation with premedication. This study confirms the feasibility and the informative value of cerebral NIRS monitoring in this setting.Clinical Trial Registration: www.ClinicalTrials.gov, identifier NCT02700893.

On accessibility for pro-p groups

We define the notion of accessibility for a pro-p group. We prove that finitely generated pro-p groups are accessible given a bound on the size of their finite subgroups. We then construct a finitely generated inaccessible pro-p group, and also a finitely generated inaccessible discrete group which is residually p-finite.

The Kummerian property and maximal pro-p Galois groups

For a prime number p, we give a new restriction on pro-p groups G which are realizable as the maximal pro-p Galois group GF(p) for a field F containing a root of unity of order p. This restriction arises from Kummer Theory and the structure of the maximal p-radical extension of F. We study it in the abstract context of pro-p groups G with a continuous homomorphism θ:G→1+pZp, and characterize it cohomologically, and in terms of 1-cocycles on G. This is used to produce new examples of pro-p groups which do not occur as maximal pro-p Galois groups of fields as above.

Explicit ring-theoretic presentation of Iwasawa algebras

Iwasawa algebras are completed group algebras of compact p-adic Lie groups. Ardakov and Venjakob have studied the structure theory and the ring-theoretic properties of such algebras. This article gives an explicit presentation by generators and relations of the Iwasawa algebras of uniform pro-p groups, i.e. the pro-p groups that admit a p-adic analytic manifold structure.

Maximal deviation of large powers in the Nottingham group

Abstract The group of formal power series under substitution over 𝔽 p {\mathbb{F}_{p}} , the so-called Nottingham group, is a pro-p group so it is a metric space. The intertwining of these two structures is an important object of study. This paper is concerned with how the power map influences the distance of elements. Suppose that d ⁢ ( f , g ) = d 1 {d(f,\,g)=d_{1}} while d ⁢ ( f , 1 ) = d 2 ≥ d 1 {d(f,1)=d_{2}\geq d_{1}} . In this paper we provide a sharp bound for d ⁢ ( f j , g j ) {d(f^{j},\,g^{j})} in terms of d 1 , d 2 {d_{1},\,d_{2}} and the exponent j. This bound confirms a conjecture by K. Keating.

On small waist pairs in pro-p groups

In this paper we analyze a list of general properties of waists and waist pairs in a pro-p group, these being subgroups or pairs of subgroups comparable in some sense with respect to inclusion with any open normal subgroup of the pro-p group. In the case of waist pairs of a pro-p group we get some characterizations of them, and give a way of building new waist pairs.

Novel combination techniques for anaesthetic induction to manage perioperative hypothermia: A prospective, randomized, controlled, single blind trial

Background: Intraoperative hypothermia is common issue with serious consequences occurred during anaesthesia. However, less attention has been directed to preventing redistribution hypothermia there is need of effective techniques to develop. Objective: In this study, we compared three different anaesthetic induction techniques to standard IV propofol inductions (control) in their effect on reducing redistribution hypothermia. Methods: Elective, afebrile patients, age 18 to 57 years, were randomly assigned to one of four groups (n = 60 each). Group “PROP” was induced with 2.2 mg/kg propofol, Group “INH/100” with 8% sevoflurane in 100% oxygen, Group “INH/50” with 8% sevoflurane in 50% oxygen and 50% nitrous oxide, and Group “Phnl/PROP” with 2.2 mg/kg propofol immediately preceded by 160 mcg phenylephrine. Patients were maintained with sevoflurane in 50% nitrous oxide and 50% oxygen in addition to opioid narcotic. Forced air warming was used. Core temperatures were recorded every 15 min after induction for 1 h. Results: Compared to control group PROP, the mean temperatures in groups INH/100, INH/50, and Phnl/PROP were higher 15, 30, 45 and 60 min after induction, averaging between 0.29 °C and 1.0 °C higher (p< 0.001 for all comparisons). There were statistically significant differences in the mean temperatures between groups INH/ 100 and INH/50, INH/100 and Phyl/PROP, and INH/50 and Phyl/PROP at any time point (all p< 0.05). Few patients in three groups had a core temperature >37.5 °C at T60 time point, except PROP group. Conclusions: The inhalation inductions with sevoflurane or with prophylactic phenylephrine bolus prior to propofol induction reduced the magnitude of redistribution hypothermia by an average of 0.29 to 1.0 °C in patients aged 18 to 57 years.

Prime decomposition and the Iwasawa MU-invariant

AbstractFor Γ = ℤp, Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large μ-invariants. In this work, we investigate other uniform pro-p groups which are realisable as Galois groups of towers of number fields with arbitrarily large μ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p−1. Both in Iwasawa's work, and in the present one, the size of the μ-invariant appears to be intimately related to the existence of primes that split completely in the tower.

On the Invariant Factors of Class Groups in Towers of Number Fields

AbstractFor a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Actions of rigid groups on UHF-algebras

Let Λ be a countably infinite property (T) group, and let D be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of Λ on D. The proof consists in assigning, to any second countable abelian pro-p group G, a strongly outer action of Λ on D whose (weak) cocycle conjugacy class completely remembers the group G. The group G is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function.Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of Λ on D are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when Λ is only assumed to contain an infinite subgroup with relative property (T), and for actions on (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebras with at least one trace.Finally, we use the techniques of this paper to construct outer actions on R with prescribed cohomology. Precisely, for every infinite property (T) group Λ, and for every countable abelian group Γ, we construct an outer action of Λ on R whose 1-cohomology is isomorphic to Γ.

Cohomology of uniserial p-adic space groups with cyclic point group

Let p be a fixed prime number and let R denote a uniserial p-adic space group of dimension dx=px−1(p−1) and with cyclic point group of order px. In this short note we prove that all the quotients of R of size bigger than or equal to pdx+x have isomorphic mod p cohomology groups. In particular, we show that the cohomology groups of sufficiently large quotients of the unique maximal class pro-p group are isomorphic as Fp-modules.

Approximation by subgroups of finite index and the Hanna Neumann conjecture

Let F be a free group (pro-p group), and let U and W be two finitely generated subgroups (closed subgroups) of F. The Strengthened Hanna Neumann conjecture says that ∑x∈U\F/Wrk¯(U∩xWx−1)≤rk¯(U)rk¯(W),whererk¯(U)=max {rk(U)−1,0}. This conjecture was proved independently in the case of abstract groups by J. Friedman and I. Mineyev in 2011. In this paper we give the proof of the conjecture in the pro-p context, and we present a new proof in the abstract case. We also show that the Luck approximation conjecture holds for free groups.

Virtually free pro-p products

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).

Test elements in pro-p groups with applications in discrete groups

Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ϕ : G → G, ϕ(g) = g implies ϕ is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.

The torsion-free rank of homology in towers of soluble pro-p groups

We show that for every finitely presented pro-p nilpotent-by-abelian-by-finite group G there is an upper bound on $${\dim _{{\mathbb{Q}_p}}}\left( {{H_1}\left( {M,{\mathbb{Z}_p}} \right){ \otimes _{{\mathbb{Z}_p}}}{\mathbb{Q}_p}} \right)$$ , as M runs through all pro-p subgroups of finite index in G.

P-Johnson homomorphisms and pro-p groups

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Zp-extension of a number field. We give their cohomological interpretation in terms of Massey products in Galois cohomology.

The pro-p group of upper unitriangular matrices

We study the pro-p group G whose finite quotients give the prototypical Sylow p-subgroup of the general linear groups over a finite field of prime characteristic p. In this article, we extend the known results on the subgroup structure of G. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski [13–15] studies a free product Cp⁎Cp as a (discrete) subgroup of G and we prove that its closure is selfnormalising of infinite index in the subgroup of 2-periodic elements of G. We also discuss change of rings: field extensions and a variant for the p-adic integers, this latter linking G with some well known p-adic analytic groups. Finally, we calculate the Hausdorff dimensions of some closed subgroups of G and show that the Hausdorff spectrum of G is the whole interval [0,1] which is obtained by considering partition subgroups only.

Sur la propagation de la propriété mild au-dessus d’une extension quadratique imaginaire de $$\varvec{\mathbb {Q}}$$ Q

In this work, we are interested in the pro-p groups \(G_S\), which are Galois groups of maximal pro-p extensions of number fields unramified outside a finite set S of primes not dividing p. We focus on whether the mildness property is preserved over imaginary quadratic extensions. Our starting point is Labute-Schmidt’s criterion (Schmidt, Doc Math 12:441–471, 2007), based on the study of the cup-product on the first cohomology group \(H^1(G_S,\mathbb {F}_p)\). In favourable conditions, we show by computation that the group we study often satisfies a weak version (\(LS_f\)) of Labute-Schmidt’s criterion. Then, a theoretical criterion is established for proving mildness of some groups to which the (\(LS_f\)) criterion does not apply. This theoretical criterion is finally illustrated by examples for \(p=3\) and compared to Labute and Vogel’s works (Labute, J Reine Angew Math 596:155–182, 2006 et Vogel, Circular sets of primes of imaginary quadratic number fields, 2006).

Ascending chains of finitely generated subgroups

We show that a nonempty family of n-generated subgroups of a pro-p group has a maximal element. This suggests that ‘Noetherian Induction’ can be used to discover new features of finitely generated subgroups of pro-p groups. To demonstrate this, we show that in various pro-p groups Γ (e.g. free pro-p groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup H≠1 is the greatest subgroup of Γ containing H as an open subgroup. We also show that an ascending chain of n-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich–Myasnikov).