Free Abelian Group Research Articles

A counterexample to the easy direction of the geometric Gersten conjecture.

For finitely generated groups H and G, equipped with word metrics, a translation-like action of H on G is a free action such that each element of H acts by a map which has finite distance from the identity map in the uniform metric. For example, if H is a subgroup of G, then right translation by elements of H yields a translation-like action of H on G. Whyte asked whether a group with no translation-like action by a Baumslag-Solitar group must be hyperbolic, where the free abelian group of rank 2 is understood to be a Baumslag-Solitar group. We show that the converse of this conjecture is false, and in particular the fundamental group of a closed hyperbolic 3-manifold admits a translation-like action by the free abelian group of rank 2.

On the topological mass lattice groups

Let G to be a torsion free abelian group. In this paper we introduce the following concepts: Some interesting results about unital lattice groups and Riesz spaces with the chief link topology on them have been presented.

On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

AbstractThis paper explores the structure groupsG(X,r)of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient$\overline {G}_{(X,r)}$ofG(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: ifXinjects intoG(X,r), then it also injects into$\overline {G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization ofG(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.

EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO

We consider the following question: for which metrizable separable spaces does the free abelian topological group isomorphically embed into . While for many natural spaces such an embedding exists, our main result shows that if is a Cook continuum or is a rigid Bernstein set, then does not embed into as a topological subgroup. The analogous statement is true for the free boolean group .

The Mackey problem for free abelian topological groups

The classical Mackey–Arens theorem states that every locally convex space has a Mackey space topology. However, in the wider class of locally quasi-convex (lqc) groups an analogous result does not hold. Indeed, Ausenhofer and the author showed independently that the free abelian topological group $$A(\mathbf {s})$$ over a convergent sequence $$\mathbf {s}$$ does not admit a Mackey group topology. We essentially extend this example by showing that the free abelian topological group A(X) over a non-discrete zero-dimensional (for example, countable) metrizable space X does not have a Mackey group topology.

Crystal monoids & crystal bases: Rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2

The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time.

Simple groups separated by finiteness properties

We show that for every positive integer n there exists a simple group that is of type $$\mathrm {F}_{n-1}$$ but not of type $$\mathrm {F}_n$$ . For $$n\ge 3$$ these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace–Remy, consists of non-affine Kac–Moody groups over finite fields. Our examples arise from Rover–Nekrashevych groups, and contain free abelian groups of infinite rank.

Intersection problem for Droms RAAGs

We solve the subgroup intersection problem (SIP) for any RAAG [Formula: see text] of Droms type (i.e. with defining graph not containing induced squares or paths of length [Formula: see text]): there is an algorithm which, given finite sets of generators for two subgroups [Formula: see text], decides whether [Formula: see text] is finitely generated or not, and, in the affirmative case, it computes a set of generators for [Formula: see text]. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. [Formula: see text]) even have unsolvable SIP.

Free products and the algebraic structure of diffeomorphism groups

Let $M$ be a compact one--manifold, and let $\mathrm{Diff}^{1+\mathrm{bv}}(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation. We prove that if $G$ is a group which is not virtually metabelian, then $(G\times\mathbb{Z})*\mathbb{Z}$ is not realized as a subgroup of $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. This gives the first examples of finitely generated groups $G,H\le \mathrm{Diff}_+^\infty(M)$ such that $G\ast H$ does not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. By contrast, for all countable groups $G,H\le\mathrm{Homeo}^+(M)$ there exists an embedding $G\ast H\to \mathrm{Homeo}^+(M)$. We deduce that many common groups of homeomorphisms do not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$, for example the free product of $\mathbb{Z}$ with Thompson's group $F$. We also complete the classification of right-angled Artin groups which can act smoothly on $M$ and in particular, recover the main result of a joint work of the authors with Baik. Namely, a right-angled Artin group $A(\Gamma)$ either admits a faithful $C^{\infty}$ action on $M$, or $A(\Gamma)$ admits no faithful $C^{1+\mathrm{bv}}$ action on $M$. In the former case, $A(\Gamma)\cong\prod_i G_i$ where $G_i$ is a free product of free abelian groups. Finally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.

An algebraic description of screw dislocations in SC and BCC crystal lattices

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately.

Whitehead’s problem and reverse mathematics

Whitehead’s Problem asks whether every Whitehead group is a free abelian group. Stein in 1951 showed that every countable Whitehead group is a free abelian group, but Shelah in 1974 showed that Whitehead’s Problem in general is independent of ZFC. In this paper, we study Whitehead’s Problem in reverse mathematics and, in particular, the proof-theoretic strength of Stein’s Theorem. We establish that over the base theory WKL0, ACA0 is equivalent to Stein’s Theorem, whereas over the base theory RCA0, WKL0 and Stein’s Theorem are incomparable; this is done by showing that Stein’s Theorem is true in REC.

The Algebraic Structure of Quantity Calculus

Abstract The algebraic structure underlying the quantity calculus is defined axiomatically as an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product, and quotient spaces are considered, as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation of quantity calculus which is centered on the concept of dimension, rather than on the concept of unit, which is regarded as secondary, and uses only integer exponents of the dimensions.

On Bellman’s and Knuth’s Problems and their Generalizations

Various generalizations of the classical problem of the fastest raising to a power (or the so-called problem on addition chains) are studied in the asymptotic sense. Under weak restrictions, we demonstrate asymptotically tight solutions of the two best known generalizations, namely, Bellman’s problem on the computational complexity (on the minimal number of multiplication operations) of a normed monomial of several variables and Knuth’s problem on the computational complexity of a power system of one variable. We also briefly review some results on the computational complexity for three problems, namely, the computation of p-element systems of normed monomials in q variables, additive computations for systems of p integer linear forms over q variables, and the computation of p-element systems of the free Abelian group with q generators.

Alexander invariants of ribbon tangles and planar algebras

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, bounding $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, throught welded diagrams. We give a simple combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. When restricted to diagrams without virtual crossings, $\mathsf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author.

On the SL(2, C)-representation rings of free abelian groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

Towers of regular self-covers and linear endomorphisms of tori

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that $\pi_1(M)$ surjects onto a nontrivial free abelian group $A$, and the self-cover is induced by a linear endomorphism of $A$. Under further hypotheses we show that a finite cover of $M$ admits the structure of a principal torus bundle. We show that this applies when $M$ is K\"ahler and $p$ is a strongly regular, holomorphic self-cover, and prove that a finite cover splits as a product with a torus factor.

Construction of non-commutative surfaces with exceptional collections of length 4

Recently de Thanhoffer de V\"olcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to $\mathbb{P}^1\times\mathbb{P}^1$ (and noncommutative quadrics), and an infinite family indexed by the natural numbers. For $m=0,1$ there are commutative and noncommutative surfaces having this Euler form, whilst for $m\geq 2$ there are no commutative surfaces. In this paper we construct sheaves of maximal orders on surfaces having these Euler forms, giving a geometric construction for their numerical blowups.

Note on the question of Sikora

A natural topology on the set of left orderings on free abelian groups and free groups [Formula: see text], [Formula: see text] has studied in [A. S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36(4) (2004) 519–526; L. Smith, On ordering free groups, J. Symbolic Comput. 40 (2005) 1285–1290, Corrigendum (with A. Clay) 44 (2009) 1529–1532]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a conjecture: this is also true for the free group [Formula: see text] with [Formula: see text] generators. We point out the paper dealing with equivalent questions.

The power graph of a torsion-free group

The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if $y$ is a power of $x$. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, we consider power graphs of torsion-free groups. Our main results are that, for torsion-free nilpotent groups of class at most $2$, and for groups in which every non-identity element lies in a unique maximal cyclic subgroup, the power graph determines the directed power graph up to isomorphism. For specific groups such as $\mathbb{Z}$ and $\mathbb{Q}$, we obtain more precise results. Any isomorphism $P(\mathbb{Z})\to P(G)$ preserves orientation, so induces an isomorphism between directed power graphs; in the case of $\mathbb{Q}$, the orientations are either all preserved or all reversed. We also obtain results about groups in which every element is contained in a unique maximal cyclic subgroup (this class includes the free and free abelian groups), and about subgroups of the additive group of $\mathbb{Q}$ and about $\mathbb{Q}^n$.

On the non-existence of the Mackey topology for locally quasi-convex groups

Abstract We give a self-contained proof that the Graev free abelian group over a convergent sequence has no Mackey topology. This means the supremum of all locally quasi-convex compatible topologies of this group is no longer compatible.