We show that the Hausdorff dimension of the closure of the second Grigorchuk group is 43/128. Furthermore, we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normalizer in the full automorphism group of the tree.

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that Nw(1) ≥ | G| k-1, where for g∈ G, the quantity Nw(g) is the number of k-tuples (g1, … , gk) ∈ G(k) such that w(g1, … , gk) = g. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that Nw(g) ≥ | G| k-

For a generalized (or non-central) camera model, the minimal problem for two views of six points has efficient solvers. However, minimal problems of three views with four points and three views of six lines have not yet been explored and solved, despite the efforts from the computer vision community. This paper develops the formulations of these two minimal problems and shows how state-of-the-art

Research connecting text and images has recently seen several breakthroughs, with models like CLIP, DALL•E 2, and Stable Diffusion. However, the connection between text and other visual modalities, such as lidar data, has received less attention, prohibited by the lack of text-lidar datasets. In this work, we propose LidarCLIP, a mapping from automotive point clouds to a pre-existing CLIP embeddin

The following problem was originally posed by B. H. Neumann and H. Neumann.Suppose that a group can be generated by elements and that is a homomor-phic image of . Does there exist, for every generating -tuple (ℎ1, … , ℎ ) of ,a homomorphism ∶ → and a generating -tuple (1, … , ) of such that( 1 , … , ) = (ℎ1, … , ℎ )?M. J. Dunwoody gave a negative answer to this question, by means of a carefullyeng

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