Linear Bounds for the Degree of Subgroup Growth in Terms of the Hirsch Length

Let G be a residually-finite virtually soluble minimax group.We give upper and lower linear bounds for the degree of subgroupgrowth of G in terms of the Hirsch length of G. 2000 MathematicsSubject Classification 20F99.     

Gegen Loten abgeschlossene Untergruppen metrischer Vektorräume

The purpose of this paper is to describe for any finite dimensional anisotropic metric vector space V=V(K,f) over a field K all those subgroups U of the additive group of V satisfying the condition: If f(a,b)=0 and a+b U then a,b U, for all a,b V.     

Arithmetic Groups,Base Change,and Representation Growth

Consider an arithmetic group \({\mathbf{G}(O_S)}\), where \({\mathbf{G}}\) is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers O _S of a number field K with respect to a finite set of places S. For each \({n \in \mathbb{N}}\), let \({R_n(\mathbf{G}(O_S))}\) denote the number of irreducible complex representations of \({\mathbf{G}(O_S)}\) of dimension at most n. The degree of representation growth \({\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty}\log R_n(\mathbf{G}(O_S)) / \log n}\) is finite if and only if \({\mathbf{G}(O_S)}\) has the weak Congruence Subgroup Property. We establish that for every \({\mathbf{G}(O_S)}\) with the weak Congruence Subgroup Property the invariant \({\alpha(\mathbf{G}(O_S))}\) is already determined by the absolute root system of \({\mathbf{G}}\). To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions \({K{\subset}L}\). We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines.     

Set the controls for the heart of the Sun

In this paper we describe experiments conducted with high-power lasers that are attempting to replicate, for a very short time and in miniature, conditions found in the Sun. Experiments to date have reached conditions in the outer part of the Sun. To reach the Sun's centre requires compression of material to very much greater than solid density and heating to over ten million degrees. To achieve this, a new class of experiments and a new generation of high-power lasers are required.     

The Magnus property for direct products

A group G is said to have the Magnus property if the following holds: whenever two elements x, y have the same normal closure, then x is conjugate to y or to y^?1. We prove: let p be an odd prime, and let G, H be residually finite-p groups with the Magnus property. Then the direct product \({G \times H}\) has the Magnus property. By considering suitable crystallographic groups, we give an explicit example of finitely generated, torsion-free, residually finite groups G, H with the Magnus property such that the direct product \({G \times H}\) does not have the Magnus property.     

Zeta functions of three-dimensional <Emphasis Type="Italic">p</Emphasis>-adic Lie algebras

We give an explicit formula for the subalgebra zeta function of a general three-dimensional Lie algebra over the p-adic integers . To this end, we associate to such a Lie algebra a ternary quadratic form over . The formula for the zeta function is given in terms of Igusa’s local zeta function associated to this form. We acknowledge support from the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation.     

Schwingungsspektroskopische hochdruckschemata einfacher verbindungen

Infrared spectroscopic investigations show that the following schemes are valid for the triatomic ions N₃^? and NO₂^? at high pressure. (1) High pressure strongly increases the stretching force constants of a linear system like N₃^? compared with the data at normal pressure. For statistical reasons the bond angle α is unchanged, in agreement with the fact that the bending force constant f_α is unaffected within experimental accuracy. (2) Statistical considerations predict a decrease in α for bent molecules XY₂(C_2v at high pressure. For NO₂^?, this decrease is about 3° at 35 kbar, combined with a significant increase in the bending force constant f_°. Both the stretching force constant f_NO and the interaction force constant f_NO/NO decrease considerably.     

Conversion of Dinitrogen into Acetonitrile under Ambient Conditions

About 20 % of the ammonia production is used as the chemical feedstock for nitrogen‐containing chemicals. However, while synthetic nitrogen fixation at ambient conditions has had some groundbreaking contributions in recent years, progress for the direct conversion of N₂ into organic products remains limited and catalytic reactions are unknown. Herein, the rhenium‐mediated synthesis of acetonitrile using dinitrogen and ethyl triflate is presented. A synthetic cycle in three reaction steps with high individual isolated yields and recovery of the rhenium pincer starting complex is shown. The cycle comprises alkylation of a nitride that arises from N₂ splitting and subsequent imido ligand centered oxidation to nitrile via a 1‐azavinylidene (ketimido) intermediate. Different synthetic strategies for intra‐ and intermolecular imido ligand oxidation and associated metal reduction were evaluated that rely on simple proton, electron, and hydrogen‐atom transfer steps.