Finite Factor Representations of 2-Step Nilpotent Groups and the Orbit Method

2005 ◽  
Vol 131 (2) ◽  
pp. 5508-5519
Author(s):  
K. P. Kokhas
Keyword(s):  
Nilpotent Groups ◽  
Orbit Method ◽  
Finite Factor

Preservation of saturation and stability in a variety of nilpotent groups

1981 ◽  
Vol 46 (3) ◽  
pp. 499-512 ◽  
Author(s):  
Pat Rogers
Keyword(s):  
Normal Form ◽  
Free Product ◽  
Nilpotent Groups ◽  
Principal Tool ◽  
Finite Factor ◽  
Torsion Groups ◽  
Basic Ideas ◽  
Necessary And Sufficient ◽  
Free Product Of Groups ◽  
Product Of Groups

This paper is a contribution to the growing literature on the model theory of nilpotent groups. (See Baumslag and Levin [2]; Eršov [5]; Hodges [9], [10]; Mal′cev [14]; Olin [16] and Saracino [19], [20].) In it we investigate the conditions under which the free product in the variety of all nilpotent of class 2 (nil-2) groups preserves saturation and stability.It is well known that the direct product preserves both saturation (see Waszkiewicz and Wȩglorz [23]) and stability (see Wierzejewski [24]; Macintyre [13]; Eklof and Fisher [4]). On the other hand it is easy to show that the full free product of groups preserves neither property; indeed, in the case of saturation this failure is extremely bad since no free product of nontrivial groups is even 2-saturated. Our results show that the nil-2 free product falls between these two extremes.Our proofs are mainly model-theoretic with a smattering of elementary algebra and rely heavily upon the unique normal form for the elements of a nil-2 free product given by MacHenry in [12]. (This normal form and some of its consequences are discussed in §1.) We assume familiarity with the basic ideas of saturation (see Chapter 5 of [3]) and Shelah's treatment of stability in [22].We prove two main theorems in §3 each giving a necessary and sufficient condition in separate situations for the preservation of saturation. In the first (Theorem 3.1) we allow one finite factor, while in the second (Theorem 3.10) we deal solely with torsion groups. Our motivation for the proof of sufficiency was the paper of Waszkiewicz and Wȩglorz [23] and the principal tool is a “Feferman-Vaught” Theorem for the nil-2 free product which we prove in §2. We also show that if both factors in a nil-2 free product are nontorsion and one factor has a nil-2 basis, then the group is not even 3-saturated. We leave open the case where both factors are infinite but only one is torsion.

Equivariant prequantization and admissible coadjoint orbits

1993 ◽  
Vol 114 (1) ◽  
pp. 131-142
Author(s):  
P. L. Robinson
Keyword(s):  
Geometric Quantization ◽  
Classical Mechanics ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Coadjoint Orbits ◽  
Type I ◽  
Quantization Scheme ◽  
Orbit Method ◽  
Irreducible Unitary Representations ◽  
Simply Connected

The orbit method has as its primary goal the construction and parametrization of the irreducible unitary representations of a (simply-connected) Lie group in terms of its coadjoint orbits. This goal was achieved with complete success for nilpotent groups by Kirillov[8] and for type I solvable groups by Auslander and Kostant[l] but is known to encounter difficulties when faced with more general groups. Geometric quantization can be viewed as an outgrowth of the orbit method aimed at providing a geometric passage from classical mechanics to quantum mechanics. Whereas the original geometric quantization scheme due to Kostant[9] and Souriau[14] enabled such a passage in a variety of situations, it too encounters difficulties in broader contexts.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Powerfully nilpotent groups of rank 2 or small order

2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams
Keyword(s):  
Detailed Analysis ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Rank 2 ◽  
Full Classification

AbstractIn this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order {p^{n}}. We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to {p^{6}}.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

scholarly journals Equations in nilpotent groups

2015 ◽  
Vol 143 (11) ◽  
pp. 4723-4731 ◽  
Author(s):  
Moon Duchin ◽  
Hao Liang ◽  
Michael Shapiro
Keyword(s):  
Nilpotent Groups
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