Gamma—rings and multiplications on abelian groups

If M and Γ are abelian groups, then M will be a Γ-ring iff there exists a group homomorphism f from Γ into the group of all multiplications of M, Mult(M), such that f(Γ) satisfies the Generalized Associativity Property on M. In this note we examine the following special cases of this result: (i) M is a Γ-ring satisfying the Nobusawa Condition, (ii) M is a cyclic group, (iii) M is a direct sum of cyclic groups and (iv) M is a Γ-ring that has unity elements.     

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

Geometric analysis on quaternion ℍ-type groups

We construct some examples of ℍ-types Carnot groups related to quaternion numbers and study their geometric properties. We involve the Hamiltonian formalism to obtain the equations of geodesics and calculate the cardinality of geodesics joining two different points on these groups. We prove Kepler’s law and give a nice geometric interpretation of the length of geodesies.     

Tur��n��s extremal problem on locally compact abelian groups

Let G be a locally compact abelian group (LCA group) and ?? be an open, 0-symmetric set. Let F:= F(??) be the set of all continuous functions f: G ?? ? which are supported in ?? and are positive definite. The Turán constant of ?? is then defined as $$ \mathcal{T}(\Omega ): = \sup \left\{ {\int_\Omega {f:f \in \mathcal{F}} (\Omega ),f(0) = 1} \right\} $$ . Mihalis Kolountzakis and the author has shown that structural properties ?? like spectrality, tiling or packing with a certain set ?? ?? of subsets ?? in finite, compact or Euclidean (i.e., ? ^d ) groups and in ? ^d yield estimates of T (??). However, in these estimates some notion of the size, i.e., density of ?? played a natural role, and thus in groups where we had no grasp of the notion, we could not accomplish such estimates. In the present work a recent generalized notion of asymptotic uniform upper density is invoked, allowing a more general investigation of the Turán constant in relation to the above structural properties. Our main result extends a result of Arestov and Berdysheva, (also obtained independently and along different lines by Kolountzakis and the author), stating that convex tiles of a Euclidean space necessarily have $$ \mathcal{T}_{\mathbb{R}^d } (\Omega )\left| \Omega \right|/2^d $$ . In our extension ? ^d could be replaced by any LCA group, convexity is considerably relaxed to ?? being a difference set, and the condition of tiling is also relaxed to a certain packing type condition and positive asymptotic uniform upper density of the set ??. Also our goal is to give a more complete account of all the related developments and history, because until now an exhaustive overview of the full background of the so-called Turán problem was not delivered.     

Fredman’s reciprocity,invariants of abelian groups,and the permanent of the Cayley table

Let C_n{\mathcal{C}}_{n} denote the cyclic group of order n. For G=C_nG={\mathcal{C}}_{n}, we compute the Poincaré series of all C_n{\mathcal{C}}_{n}-isotypic components in (the symmetric tensor exterior algebra of ). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili–Jibladze. Then we consider the Cayley table, , of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of equals , where n is the order of G.     

On Σ-subsets of naturals over abelian groups

We obtain conditions for the Σ-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each e-ideal I there exists a torsion-free abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I; there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal Σ-function; for each principal e-ideal I there exists a periodic abelian group A such that the family of e-degrees of Σ-subsets of ω in $\mathbb{H}\mathbb{F}(A)$ coincides with I.     

Torsion-free abelian α-irreducible groups of finite rank

If F is a free abelian group of finite rank and α is an endomorphism or an automorphism of its divisible hull, then the α‐ hull is determined, i.e. the minimal torsion-free abelian group with this endomorphism a. Torsion-free abelian groups of finite rank are called α-irreducible if their divisible hull is α-irreducible for an automorphism a. A complete classification is given for α-irreducible groups and this result is applied to groups of rank 2.     

A Szemerédi-type regularity lemma in abelian groups, with applications

Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorem for sets which are almost sum-free. If has δ N² triples (a₁, a₂, a₃) for which a₁ + a₂ = a₃ then A = B ∪ C, where B is sum-free and |C| = δ′N, and as Another answers a question of Bergelson, Host and Kra. If 0,$$" align="middle" border="0"> if \,N_{0}(\alpha, \epsilon)$$" align="middle" border="0"> and if has size α N, then there is some d ≠ 0 such that A contains at least three-term arithmetic progressions with common difference d.Received: November 2003 Revision: October 2004 Accepted: December 2004     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

Quasi-K?hler groups, 3-manifold groups, and formality

In this note, we address the following question: Which 1-formal groups occur as fundamental groups of both quasi-K?hler manifolds and closed, connected, orientable 3-manifolds. We classify all such groups, at the level of Malcev completions, and compute their coranks. Dropping the assumption on realizability by 3-manifolds, we show that the corank equals the isotropy index of the cup-product map in degree one. Finally, we examine the formality properties of smooth affine surfaces and quasi-homogeneous isolated surface singularities. In the latter case, we describe explicitly the positive-dimensional components of the first characteristic variety for the associated singularity link.     

Almost block independence and bernoullicity of ℤ d -actions by automorphisms of compact abelian groups

Summary We prove that a ^d -action by automorphisms of a compact, abelian group is Bernoulli if and only if it has completely positive entropy. The key ingredients of the proof are the extension of certain notions of asymptotic block independence from -actions to ^d -action and their equivalence with Bernoullicity, and a surprisingly close link between one of these asymptotic block independence properties for ^d -actions by automorphisms of compact, abelian groups and the product formula for valuations on global fields.Oblatum 20-X-1994     

A theorem of Piccard’s type in abelian Polish groups

In the paper we prove a theorem of Piccard’s type which generalizes [9, Theorem 2]. More precisely, we show that in an abelian Polish group X the set \(\left\{ {\left( {{x_{1, \ldots ,\;}}{x_N}} \right) \in \;{X^N}\;:\;A\; \cap \;\bigcap\limits_{i = 1}^N {\left( {A + {x_i}} \right)} \;is\;not\;Haar\;meager\;in\;X} \right\}\) is a neighbourhood of 0 for every N ∈ N and every Borel non-Haar meager set A ? X. The paper refers to the paper [3].     

When must pure extensions of countable abelian groups necessarily split?

Necessary and sufficient conditions are given for the group of pure extensions of a countable abelian group by a countable abelian group to equal zero.

     

Fourier–Mukai transform on abelian surfaces

We study moduli spaces of stable sheaves on abelian surfaces whose Mukai vectors are related by a cohomological Fourier-Mukai transform. We show that there is a Fourier-Mukai transform inducing a birational map between them.     

Bridgeland’s stabilities on abelian surfaces

In this paper, we shall study the structure of walls for Bridgeland’s stability conditions on abelian surfaces. In particular, we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the Picard number 1.