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Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum here hdim p G : (X|X ⊂ G) [0,1]denotes the Hausdorff dimension function associated to the p-power series p:Gpi, i ε N0 More precisely, we show that hspec p (G)=[0,1/3] ∪ (1) contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff s
Let G be a finitely generated pro-p group, equipped with the p-power series P:Gi=GPi, i ∈ ℕ0. The associated metric and Hausdorff dimension function hdimGP:{X|X⊆G}→[0,1] give rise to hspecP(G)={hdimGP(H)|H≤G}⊆[0,1], the Hausdorff spectrum of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, hspecP(G) consists of finit
A multi-edge spinal group is a subgroup of the automorphism group of a regular p-adic rooted tree, generated by one rooted automorphism and a finite number of directed automorphisms sharing a common directing path. We prove that torsion multi-edge spinal groups do not have maximal subgroups of infinite index. This generalizes a result of Pervova for GGS-groups.
Let G be a finite group. We show that the order of the subgroup generated by coprime γk-commutators (respectively,δk- commutators) is bounded in terms of the size of the set of coprime γk-commutators (respectively, δk-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words γk and δk are concise.
A group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate p k -central groups for p a prime number. For G a finite group of exponent pk , the covering group of G is pk -central. Using this we show that the exponent of the Schur multiplier of G is bounded by p⌈c/p-1⌉, where c is the nilpotency class of G. Next we give an explic
In a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of But
A p-group G is p-central if Gp ≤ Z(G), and G is p2-abelian if (xy)p2 = xp2 yp2 for all x; y ε G. We prove that for G a finite p2-abelian p-central p-group, excluding certain cases, the order of G divides the order of Aut(G).
We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large, which implies that Schlage-Puchta's infinite finitely generated p-groups are not finitely presented. We also show that for all primes p at least 7, any group having a presentation of p-deficiency greater than 1 is GolodS
A global energy transition requires alternatives to fossil fuels in energy-intensive industries and transport sectors, which are particularly reliant on the unique material properties of fossil fuels as fuel and as feedstock. Renewable energy transitions, therefore, demand large-scale investments in green hydrogen to produce substitutes as a means of indirect electrification. In the context of Eur
This article provides some brief reflections on the ideas raised in the special issue on the social dimension of EU trade policies from a normative power (NP) perspective. A NP perspective attempts to understand and judge the ideational aspects of the EU by studying the EU’s principles, actions and impact in world politics. The article first elaborates on the role of labour rights as human rights
The mythology of the European Union (EU) in world politics can be told and untold in many different ways. This article focuses on the lore or stories of who did what to whom, the ideological projection of the past onto the present and the escapist pleasure of storytelling in looking at the mythology of ‘global Europa’– the EU in the world. It concludes with a reflection on the way in which the man
The role of symbolism in European integration provides one way of answering Craig Calhoun's 2003 call in Comparative European Politics for a means of transcending specific regimes of analysis in order to advance European studies. The article argues that our understanding of the integration process and the constitution of the European Union (EU) is furthered by broadly studying symbolic forms in a
The European Union is often compared to other political systems in order to better understand its basic features and how they structure politics. This article argues that this focus on comparative politics instils a domestic bias into the study of the EU, which also ignores the impact of enlargement. To remedy this, a comparison is suggested between the order of the EU as a regional international
Boken skall– dels ge en introduktion till den unika rättsordning inom folkrätten, som Nordatlantiska fördragsorganisationen – Nato – och dess verksamhet grundar sig på och som benämns Natorätt,– dels förklara hur Sverige påverkas av den som part i nordatlantiska fördraget (SÖ 2024:1).The book- provides an introduction to the unique legal regime in international law on which the North Atlantic Treaty Organization - NATO - and its activities are based, known as NATO law, and- explains how Sweden is affected by it as a party to the North Atlantic Treaty.
The themes of this workshop on EU civilian and military power mirror nicely the three aspects of normative power Europe that I set out to interrogate over five years ago (see a first consideration in Manners 2000a). As the above abstract from my JCMS article makes clear, I argued that previous theoretical accounts of the EU had underplayed the ontological conceptualisation of the EU as a changer o
