Free Abelian Group Research Articles

Finitely Presented Kernels of Homomorphisms From Hyperbolic Groups Onto Free Abelian Groups

Finitely Presented Kernels of Homomorphisms From Hyperbolic Groups Onto Free Abelian Groups

Galois theory and homology in quasi-abelian functor categories

Given a finite category T, we consider the functor category AT, where A can be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories AT for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of AT whose objects are given by the functors F:T→A with F(T)=0 for all T∉S. We prove that F is a torsion-free Birkhoff subcategory of AT. This allows us to study (higher) central extensions from categorical Galois theory in AT with respect to F and generalized Hopf formulae for homology.

Centralisers of linear growth automorphisms of free groups

Abstract In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

Schur rings over free Abelian group of rank two

Schur rings are subrings of the group ring of the group G afforded by a partition of G into finite sets. In this paper, Schur rings over a free abelian group of rank two are classified under the assumption that one of the direct factors is a union of some basic sets. There are eight different types, and all but one type of which are traditional.

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups

We study a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the group ring of finitely generated abelian groups. The Batalin-Vilkovisky algebra structure for finite abelian groups comes from the fact that the group ring of finite groups is a symmetric algebra, and the Batalin-Vilkovisky algebra structure for free abelian groups of finite rank comes from the fact that its group ring is a Calabi-Yau algebra.

مونويدات جزئية من الزمر الباراتبولوجية الآبيلية

One of the important subclasses of abelian paratopological groups is called the free abelian paratopological group on a topological space. It was introduced in 2003 by Remaguara, Sanchis, and Tkachenko. In this paper, we introduce a single submonoid of the free abelian paratopological group on Alexandroff space, then we prove that this submonoid is a base at the identity element of the free topology of the abelian group. The main result of this paper is to give applications of this submonoid such as studying separation axioms, compactness, and other properties of the free topology on the abelian group. Keywords: free group, Abelian paratopological group, free abelian paratopological group, Alexandroff space, neighborhood base at an identity.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Equivariant lattice bases

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in Z ( N × [ c ] ) \mathbb {Z}^{(\mathbb {N}\times [c])} , where c ∈ N c\in \mathbb {N} , has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.

Topological insulators and K-theory

We analyze topological invariants, in particular Z2 invariants, which characterize time reversal invariant topological insulators, in the framework of index theory and K-theory. After giving a careful study of the underlying geometry and K-theory, we formalize topological invariants as elements of KR theory. To be precise, the strong topological invariants lie in the higher KR groups of spheres; KR̃−j−1(SD+1,d). Here j is a KR-cycle index, as well as an index counting off the Altland-Zirnbauer classification of Time Reversal Symmetry (TRS) and Particle Hole Symmetry (PHS)—as we show. In this setting, the computation of the invariants can be seen as the evaluation of the natural pairing between KR-cycles and KR-classes. This fits with topological and analytical index computations as well as with Poincaré Duality and the Baum–Connes isomorphism for free Abelian groups. We provide an introduction starting from the basic objects of real, complex and quaternionic structures which are the mathematical objects corresponding to TRS and PHS. We furthermore detail the relevant bundles and K-theories (Real and Quaternionic) that lead to the classification as well as the topological setting for the base spaces.

Homotopy type through homology groups

We show that for certain pairs of nonnegative integers, if the reduced integral homology groups of a space are nonzero only in that pair of degrees and those groups are finitely generated free abelian groups, then the space must have the weak homotopy type of a wedge of spheres of whose dimensions belong to the pair.

Self-covering, finiteness, and fibering over a circle

A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold M M with free abelian fundamental group fibers over a circle under mild assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group Z \mathbb Z is a fiber bundle over S 1 S^1 , except for the 4 4 -dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with nonfree abelian fundamental group, which are not fiber bundles over S 1 S^1 .

Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups

Abstract Given a group $G$ and an integer $n\geq 0$ , we consider the family ${\mathcal F}_n$ of all virtually abelian subgroups of $G$ of $\textrm{rank}$ at most $n$ . In this article, we prove that for each $n\ge 2$ the Bredon cohomology, with respect to the family ${\mathcal F}_n$ , of a free abelian group with $\textrm{rank}$ $k \gt n$ is nontrivial in dimension $k+n$ ; this answers a question of Corob Cook et al. (Homology Homotopy Appl. 19(2) (2017), 83–87, Question 2.7). As an application, we compute the minimal dimension of a classifying space for the family ${\mathcal F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge 2$ . The main tools that we use are the Mayer–Vietoris sequence for Bredon cohomology, Bass–Serre theory, and the Lück–Weiermann construction.

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Vertex numbers of simplicial complexes with free abelian fundamental group

Vertex numbers of simplicial complexes with free abelian fundamental group

Intersection configurations in free and free times free-abelian groups

In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

Scalable monoids and quantity calculus

We define scalable monoids and prove their fundamental properties. Congruence relations on scalable monoids, direct and tensor products, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are defined and investigated. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. Observations relating to the theory of measurement of physical quantities accompany the results about scalable monoids. We conclude that the algebraic theory of scalable monoids and quantity spaces provides a rigorous foundation for quantity calculus.

The number of distinct cycle lengths in the cycle structure of an automorphism on an abelian group

Let G be a finitely generated free abelian group of rank n, and σ an automorphism of G. The set of distinct lengths of non-zero cycles in the cycle structure of σ is written as S(σ,G). It was proved by Klerk et al. that the structure of an arbitrary automorphism σ of G possesses at most finitely many distinct cycle lengths ([4], Proposition 3.2). A problem arises naturally: How many distinct cycle lengths possibly occur in the cycle structure of an automorphism σ of G. In this article, we prove that |S(σ,G)|≤2m if n=2m is even, or |S(σ,G)|≤2m−1 if n=2m−1 is odd, the automorphism σ for which |S(σ,G)| attains the upper bound is characterized definitely.

The group of primitive Pythagorean triples over Gaussian integers

A group structure of the set of primitive Pythagorean triples over Gaussian integers is investigated. In addition, we show that it is a free abelian group. Base on the study, the construction and enumeration of the primitive Pythagorean triples having the same hypotenuse are established.

On the Residual Finiteness of a Class of Polycyclic Groups II

Let A be a free abelian group of rank n (n ≥ 2) and ϕ be an automorphism of A, then $$A\langle \phi \rangle $$ is a polycyclic group. This paper investigates the residual finiteness of $$A\langle \phi \rangle $$ , and the special case that ϕ is a cyclic permutation of the direct components twisted by a rational integer m.

Countably compact group topologies on arbitrarily large free Abelian groups

We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.