Radial functions on free groups and a decomposition of the regular representation into irreducible components.

1981 ◽  
Vol 1981 (326) ◽  
pp. 124-135 ◽  
Keyword(s):  
Regular Representation ◽  
Free Groups ◽  
Radial Functions ◽  
Irreducible Components

scholarly journals GAUSSIAN FLUCTUATIONS OF REPRESENTATIONS OF WREATH PRODUCTS

2006 ◽  
Vol 09 (04) ◽  
pp. 529-546
Author(s):  
PIOTR ŚNIADY
Keyword(s):  
Finite Group ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Regular Representation ◽  
Wreath Products ◽  
Young Diagrams ◽  
Reducible Representation ◽  
Irreducible Components ◽  
Left Regular Representation ◽  
Left Regular

We study the asymptotics of the reducible representations of the wreath products G≀Sq = Gq ⋊ Sq for large q, where G is a fixed finite group and Sq is the symmetric group in q elements; in particular for G = ℤ/2ℤ we recover the hyperoctahedral groups. We decompose such a reducible representation of G≀Sq as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations, the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.

scholarly journals Free quotients of fundamental groups of smooth quasi-projective varieties

2021 ◽  
pp. 1-23
Author(s):  
Jose I. Cogolludo ◽  
Anatoly Libgober
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Large Rank ◽  
Small Rank ◽  
Irreducible Components ◽  
Simply Connected

Abstract We study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

scholarly journals ALGEBRAIC GEOMETRIC INVARIANTS OF PARAFREE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 155-169 ◽  
Author(s):  
SAL LIRIANO
Keyword(s):  
Algebraic Variety ◽  
Free Groups ◽  
Central Sequence ◽  
Geometric Invariants ◽  
Number Of Generators ◽  
Irreducible Components ◽  
One Relator Groups ◽  
The Difference ◽  
Representation Varieties ◽  
Trivial Consequence

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL2ℂ inherits the structure of an algebraic variety known as the representation variety of G in SL2ℂ. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between ismorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n if and only if Dim (R(G)) = 3n. In fact, a trivial consequence of Theorem 1.8 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2, there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation 1 with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups.

Sulphoamidic Phtalocyaninic Pigments and Dyes(II)

2008 ◽  
Vol 59 (5) ◽  
Author(s):  
Elena Stingaciu ◽  
Corneliu Minca ◽  
Ion Sebe
Keyword(s):  
Ir Spectra ◽  
Aromatic Amines ◽  
Free Groups ◽  
Cyanuric Chloride ◽  
Layer Chromatography ◽  
Phenylene Diamine

This work concerns the synthesis of pigments and phtalocyanine dyes obtained through the sulphonation of copper phtalocyanine and amidation with some aliphatic and aromatic amines (lauryl-amine, i-propyl-amine, hexadecyl-amine, stearyl-amine and acetyl-p-phenylene-diamine) with good properties for the electrotechnic utilisation and for toner materials. The pigments with amino free groups are transformed by condensation with cyanuric chloride in phtalocyanine pigments with different tinctorial properties. The dyes were analyzed through the layer chromatography and were characterized on the IR spectra bases and tinctorial tests.

Generating the inverse limit of free groups

2021 ◽  
Vol 578 ◽  
pp. 371-401
Author(s):  
Gregory R. Conner ◽  
Wolfgang Herfort ◽  
Curtis A. Kent ◽  
Petar Pavešić
Keyword(s):  
Inverse Limit ◽  
Free Groups

Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes

2020 ◽  
Vol 23 (4) ◽  
pp. 967-979
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Fractional Integrals ◽  
Radon Transforms ◽  
Affine Planes ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Compactly Supported ◽  
Radial Case ◽  
Compactly Supported Functions

AbstractWe apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

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 Minimal partition-free groups

2021 ◽  
Author(s):  
Afsane Bahri ◽  
Zeinab Akhlaghi ◽  
Behrooz Khosravi
Keyword(s):  
Free Groups

scholarly journals Freely indecomposable almost free groups with free abelianization

2020 ◽  
Vol 23 (3) ◽  
pp. 531-543
Author(s):  
Samuel M. Corson
Keyword(s):  
Free Groups ◽  
Uncountable Cardinals

AbstractFor certain uncountable cardinals κ, we produce a group of cardinality κ which is freely indecomposable, strongly κ-free, and whose abelianization is free abelian of rank κ. The construction takes place in Gödel’s constructible universe L. This strengthens an earlier result of Eklof and Mekler.

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