A Model-Theoretic Characterization of Countable Direct Sums of Finite Cyclic Groups #

We prove that an abelian group G is a countable direct sum of finite cyclic groups if and only if there exists a consistent existential theory Γ of abelian groups such that G is embeddable in every model of Γ.     

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L²(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II₁ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.     

Hilbertianity of division fields of commutative algebraic groups

Let G be a commutative algebraic group over a finitely generated infinite field K of characteristic p. We prove that every extension of K contained in the field obtained by adjoining to K all prime-to-p torsion points of G is Hilbertian. We also determine when the field obtained by adjoining to K all torsion points of G has this property. This extends results of Moshe Jarden on abelian varieties.     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.     

Realizations of Countable Groups as Fundamental Groups of Compacta

It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of ${\mathbb{R}^4}$ . According to the theorem of Shelah [10] such space can not be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X _G for every countable group G.     

Combings of semidirect products and 3-manifold groups

IfG is a finitely generated group that is abelian or word-hyperbolic andH is an asynchronously combable group then every split extension ofG byH is asynchronously combable. The fundamental group of any compact 3-manifold that satisfies the geometrization conjecture is asynchronously combable. Every split extension of a word-hyperbolic group by an asynchronously automatic group is asynchronously automatic.     

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ??? with respect to the set of standard generators is 3.     

Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the g-closureg_K(H) of H in K as the subgroup consisting of χ∈K such that χ(a_n)?0 in T=R/Z for every sequence {a_n} in (the Pontryagin dual of K) that converges to 0 in the topology that H induces on . We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the G_δ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Local spectral synthesis on Abelian groups

We say that a complex valued function defined on an Abelian group G is a local polynomial, if its restriction to every finitely generated subgroup of G is a polynomial. We prove that local spectral synthesis (that is, spectral synthesis using local polynomials instead of polynomials) holds on every Abelian group having countable torsion free rank. More precisely, there is a cardinal ω ₁≦κ≦2 ^ω such that local spectral synthesis holds on an Abelian group G if and only if the torsion free rank of G is less than κ.     

Simple modules for groups with abelian Sylow 2-subgroups are algebraic

An algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if G is a group with abelian Sylow 2-subgroups and K is a field of characteristic 2, then every simple KG-module is algebraic.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

Φ-Harmonic functions on discrete groups and the first ℓΦ-cohomology

We study the first cohomology groups of a countable discrete group G with coefficients in a G-module ?^Φ(G), where Φ is an N-function of class Δ₂(0) ∩ ?₂(0). Developing the ideas of Puls and Martin-Valette for a finitely generated group G, we introduce the discrete Φ-Laplacian and prove a theorem on the decomposition of the space of Φ-Dirichlet finite functions into the direct sum of the spaces of Φ-harmonic functions and ?^Φ(G) (with an appropriate factorization). We prove also that if a finitely generated group G has a finitely generated infinite amenable subgroup with infinite centralizer then \(\bar H^1\) (G, ?^Φ(G)) = 0. In conclusion, we show the triviality of the first cohomology group for the wreath product of two groups one of which is nonamenable.     

On SS-quasinormal and S-quasinormally embedded subgroups of finite groups

A subgroup H of a group G is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B. A subgroup H of a group G is said to be S-quasinormally embedded inGif for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied.     

Every finite semigroup is embeddable in a finite relatively free semigroup

The title result is proved by a Murskii-type embedding.Results on some related questions are also obtained. For instance, it is shown that every finitely generated semigroup satisfying an identity ξ^d=ξ^2d is embeddable in a relatively free semigroup satisfying such an identity, generally with a larger d; but that an uncountable semigroup may satisfy such an identity without being embeddable in any relatively free semigroup.It follows from known results that every finite group is embeddable in a finite relatively free group. It is deduced from this and the proof of the title result that a finite monoid S is embeddable by a monoid homomorphism in a finite (or arbitrary) relatively free monoid if and only if its group of invertible elements is either {e} or all of S.     

Isomorphisms of Cayley graphs of a free Abelian group

A group G is called a CI-group provided that the existence of some automorphism σ ∈ Aut(G) such that σ(A) = B follows from an isomorphism Cay(G, A) ? = Cay (G, B) between Cayley graphs, where A and B are two systems of generators for G. We prove that every finitely generated abelian group is a CI-group.     

Neighborhood Unions and Z 3-Connectivity in Graphs

Let G be a 2-edge-connected simple graph on n ≥ 14 vertices, and let A be an abelian group with the identity element 0. If a graph G* is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say that G can be A-reduced to G*. In this paper, we prove that if for every ${uv\not\in E(G), |N(u) \cup N(v)| \geq \lceil \frac{2n}{3} \rceil}$ , then G is not Z ₃-connected if and only if G can be Z ₃-reduced to one of ${\{C_3,K_4,K_4^-, L\}}$ , where L is obtained from K ₄ by adding a new vertex which is joined to two vertices of K ₄.     

Some cases of preservation of the Pontryagin dual by taking dense subgroups

We study the compact-open topology on the character group of dense subgroups of topological abelian groups. Permanence properties concerning open subgroups, quotients and products are considered. We also present some representative examples. We prove that every compact abelian group G with w(G)?c has a dense pseudocompact group which does not determine G; this provides (under CH) a negative answer to a question posed by S. Hernández, S. Macario and the third listed author two years ago.