Free Abelian Group Research Articles

On additive co-minimal pairs

A pair of non-empty subsets (W,W′) in an abelian group G is an additive complement pair if W+W′=G. The set W′ is said to be minimal to W if W+(W′∖{w′})≠G,∀w′∈W′. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. Additive complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others. The notion of minimal complements is due to Nathanson. We study tightness property of complement pairs (W,W′) such that both W and W′ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also study infinite sets forming co-minimal pairs. At the other extreme, motivated by unbounded arithmetic progressions in the integers, we look at sets which can never be a part of any minimal pair. This leads to a discussion on co-minimality, subgroups, approximate subgroups and asymptotic approximate subgroups of G.

Invariant means on Abelian groups capture complementability of Banach spaces in their second duals

Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell _\infty (G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if

Torsions and intersection forms of 4-manifolds from trisection diagrams

AbstractGay and Kirby introduced trisections, which describe any closed, oriented, smooth 4-manifold X as a union of three 4-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $\Sigma $ , guiding the gluing of the handlebodies. Any morphism $\varphi $ from $\pi _1(X)$ to a finitely generated free abelian group induces a morphism on $\pi _1(\Sigma )$ . We express the twisted homology and Reidemeister torsion of $(X;\varphi )$ in terms of the first homology of $(\Sigma ;\varphi )$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi )$ in terms of the intersection form of $(\Sigma ;\varphi )$ .

A matrix description for torsion free abelian groups of finite rank

We describe torsion free abelian groups of finite rank applying matrices with polyadic entries. This description can be considered as a modification of the classic description by A. I. Mal'cev.

Some results related to finiteness properties of groups for families of subgroups

For a group $G$ we consider the classifying space $E_{\mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{\mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of L\"uck-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube group admits a finite model for $E_{\mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $B_{\mathcal{VC}yc}(G) = E_{\mathcal{VC}yc}(G)/G$. We show for a poly-$\mathbb Z$-group $G$, that $B_{\mathcal{VC}yc}(G)$ is homotopy equivalent to a finite CW-complex if and only if $G$ is cyclic.

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

Stability of Maximum Functional Equation and Some Properties of Groups

In this research paper, we deal with the problem of determining the function χ:G→R, which is the solution to the maximum functional equation (MFE) max{χ(xy),χ(xy−1)}=χ(x)χ(y), when the domain is a discretely normed abelian group or any arbitrary group G. We also analyse the stability of the maximum functional equation max{χ(xy),χ(xy−1)}=χ(x)+χ(y) and its solutions for the function χ:G→R, where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G.

On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces

We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.

The adjoint group of a Coxeter quandle

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of $Q_W$ is isomorphic to the commutator subgroup of $W$. Finally, the root system $\Phi_W$ associated with a Coxeter group $W$ turns out to be a rack. We prove that the adjoint group of $\Phi_W$ is isomorphic to the adjoint group of $Q_W$.

Structure on the top homology and related algorithms

We explore the special structure of the top-dimensional homology of any compact triangulable space X of dimension d. Since there are no (d+1)-dimensional cells, the top homology equals the top cycles and is thus a free abelian group. There is no obvious basis, but we show that there is a canonical embedding of the top homology into a canonical free abelian group which has a natural basis up to signs. This embedding structure is an invariant of X up to homeomorphism. This circumstance gives the top homology the structure of an (orientable) matroid, where cycles in the sense of matroids correspond to the cycles in the sense of homology. This adds a novel topological invariant to the topological literature.We apply this matroid structure on the top homology to give a polynomial-time algorithm for the construction of a basis of the top homology (over Z coefficients).

Kähler groups and subdirect products of surface groups

We present a construction that produces infinite classes of K\"ahler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are K\"ahler. We apply our construction to obtain new classes of irreducible, coabelian K\"ahler subgroups of direct products of $r$ surface groups. These cover the full range of possible finiteness properties of irreducible subgroups of direct products of $r$ surface groups: For any $r\geq 3$ and $2\leq k \leq r-1$, our classes of subgroups contain K\"ahler groups that have a classifying space with finite $k$-skeleton while not having a classifying space with finitely many $(k+1)$-cells. We also address the converse question of finding constraints on K\"ahler subdirect products of surface groups and, more generally, on homomorphisms from K\"ahler groups to direct products of surface groups. We show that if a K\"ahler subdirect product of $r$ surface groups admits a classifying space with finite $k$-skeleton for $k>\frac{r}{2}$, then it is virtually the kernel of an epimorphism from a direct product of surface groups onto a free abelian group of even rank.

Quite free complicated abelian groups, pcf and black boxes

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

Spaces of knotted circles and exotic smooth structures

AbstractSuppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$ -type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic $4$ -manifolds have sets $\pi _0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi _1$ . The result about $\pi _0$ is well-known and elementary, but the result about $\pi _1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ . We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

The congruence subgroup problem for the free metabelian group on $n\geq4$ generators

The congruence subgroup problem for a finitely generated group \Gamma asks whether the map \widehat{\mathrm{Aut}(\Gamma)}\to \mathrm{Aut}(\widehat{\Gamma}) is injective, or more generally, what is its kernel C(\Gamma) ? Here \widehat{X} denotes the profinite completion of X . It is well known that for finitely generated free abelian groups C(\mathbb{Z}^{n})=\{ 1\} for every n\geq3 , but C(\mathbb{Z}^{2})=\widehat{F}_{\omega} , where \widehat{F}_{\omega} is the free profinite group on countably many generators. Considering \Phi_{n} , the free metabelian group on n generators, it was also proven that C(\Phi_{2})=\widehat{F}_{\omega} and C(\Phi_{3})\supseteq\widehat{F}_{\omega} . In this paper we prove that C(\Phi_{n}) for n\geq4 is abelian. So, while the dichotomy in the abelian case is between n=2 and n\geq3 , in the metabelian case it is between n=2,3 and n\geq4 .

Abelian quandles and quandles with abelian structure group

Sets with a self-distributive operation (in the sense of (a◃b)◃c=(a◃c)◃(b◃c)), in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang–Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either “boring” (free abelian), or “interesting” (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy (a◃b)◃c=(a◃c)◃b); present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group H2. We use this to prove that the H2 of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in H2 is important for constructing knot invariants and pointed Hopf algebras.

Long colimits of topological groups II: Free groups and vector spaces

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological R-vector space V(X) on X. We show that V(X) is a quotient of the free abelian topological group on [−1,1]×X, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of V(X) satisfy the so-called algebraic colimit property defined in the authors' previous work.

Free abelian group actions on normal projective varieties: submaximal dynamical rank case

AbstractLet X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

Dimension groups for self-similar maps and matrix representations of the cores of the associated C*-algebras

AbstractWe introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -group of the core of the C*-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ together with the unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

Metrizable quotients of free topological groups

We consider the following problem: For which Tychonoff topological spaces X do the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial (i.e. not finitely generated) metrizable quotient group? First, we give a complete solution of this problem for the key class of compact spaces X. Then, relying on this result, we resolve the problem for several more general important classes of spaces X, including the class of $$\sigma $$ -compact spaces, the class of pseudocompact spaces, the class of $$\omega $$ -bounded spaces, the class of Cech-complete spaces and the class of K-analytic spaces. Also we describe all absolutely analytic metric spaces X (in particular, completely metrizable spaces) such that the free topological group F(X) and the free abelian topological group A(X) admit a nontrivial metrizable quotient group. Our results are based on an extensive use of topological properties of non-scattered spaces.

The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight

A Bieberbach group is defined to be a torsion free crystallographic group which is an extension of a free abelian lattice group by a finite point group. This paper aims to determine a mathematical representation of a Bieberbach group with quaternion point group of order eight. Such mathematical representation is the exterior square. Mathematical method from representation theory is used to find the exterior square of this group. The exterior square of this group is found to be nonabelian. Keywords: mathematical structure; exterior square; Bieberbach group; quaternion point group