Free Abelian Group Of Rank Research Articles

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.

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.

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.

On free products inside the unit group of integral group rings

We use bicyclic units to give an explicit construction of a subgroup of isomorphic to the free product of two free abelian groups of rank two, assuming that G is a finite nilpotent group and it contains an element g of odd prime order such that the subgroup is not normal in G. To do this we first construct a subgroup isomorphic to the desired free product inside and then we find a nontrivial matrix representation of a subgroup of generated by some bicyclic units and their conjugations under the involution of We show that for an arbitrary finite group G our construction need not lead to a free product. At the end we shortly discuss possibility of constructing subgroups isomorphic to the free product of two free abelian groups of rank p − 1 for p > 3 in a similar way.

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.

Harmonic analysis on the rank- value group of a two-dimensional local field

In this work we construct a harmonic analysis on free Abelian groups of rank , namely: we construct and investigate spaces of functions and distributions, Fourier transforms and actions of discrete and extended discrete Heisenberg groups. In the case of the rank- value group of a two-dimensional local field with finite last residue field we connect this harmonic analysis with harmonic analysis on the two-dimensional local field, where the latter harmonic analysis was constructed in earlier works by the authors. Bibliography: 15 titles.

Planar pure braids on six strands

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

K-theory and K-homology of finite wreath products with free groups

Consider the wreath product $\Gamma = F\wr \mathrm{F_n} = \bigoplus_{\mathrm{F_n}}F\rtimes\mathrm{F_n}$, with $F$ a finite group and $\mathrm{F_n}$ the free group on $n$ generators. We study the Baum-Connes conjecture for this group. Our aim is to explicitly describe the Baum-Connes assembly map for $F\wr \mathrm{F_n}$. To this end, we compute the topological and the analytical K-groups and exhibit their generators. Moreover, we present a concrete 2-dimensional model for $\underline{E} \Gamma$. As a result of our K-theoretic computations, we obtain that $\mathrm K_0(\mathrm C^*_{\mathrm r}(\Gamma))$ is the free abelian group of countable rank with a basis consisting of projections in $\mathrm C^*_{\mathrm r}(\bigoplus_{\mathrm{F_n}}F)$ and $\mathrm K_1(\mathrm C^*_{\mathrm r}(\Gamma))$ is the free abelian group of rank $n$ with a basis consisting of the unitaries coming from the free group.

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.

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.

On representation varieties of free abelian groups

The reducibility of the representation variety of a free abelian group of finite rank in a semisimple non-simply connected algebraic group is proved. Irreducible components of the representation variety of a free abelian group of rank 2 in groups of type An are described.

Showing distinctness of surface links by taking 2-dimensional braids

For an oriented surface link $S$, we can take a satellite construction called a 2-dimensional braid over $S$, which is a surface link in the form of a covering over $S$. We demonstrate that 2-dimensional braids over surface links are useful for showing the distinctness of surface links. We investigate non-trivial examples of surface links with free abelian group of rank two, concluding that their link types are infinitely many.

Compact Kähler manifolds admitting large solvable groups of automorphisms

Let G be a group of automorphisms of a compact Kähler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N(G) is a free abelian group of rank ≤n−2, or G/N(G) is a free abelian group of rank n−1 and X is a complex torus, or G is a free abelian group of rank n−1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X is a projective manifold of dimension n≥3 and is not rationally connected. We also prove and use a generalization of a theorem by Fujiki and Lieberman on the structure of Aut(X).

FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

AbstractGiven a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

Fine asymptotic geometry in the Heisenberg group

For every finite generating set on the integer Heisenberg group H(Z), Pansu showed that the word metric has the large-scale structure of a Carnot-Caratheodory Finsler metric on the real Heisenberg group H(R). We study the properties of those limit metrics and obtain results about the geometry of word metrics that reflect the dependence on generators. For example we will study the probability that a group element has all of its geodesic spellings sublinearly close together, relative to the size of the element. In free abelian groups of rank at least two, that probability is 0; in infinite hyperbolic groups, the probability is 1. In H(Z) it is a rational number strictly between 0 and 1 that depends on the generating set; with respect to the standard generators, the probability is precisely 19/31.

Surface links with free abelian groups

It is known that if a classical link group is a free abelian group, then its rank is at most two. It is also known that a $k$-component 2-link group ($k$ > 1) is not free abelian. In this paper, we give examples of $T^2$-links each of whose link groups is a free abelian group of rank three or four. Concerning the $T^2$-links of rank three, we determine the triple point numbers and we see that their link types are infinitely many.

The topological decomposition of inverse limits of iterated wreath products of finite Abelian groups

Abstract. Let be an arbitrary infinite sequence of nontrivial finite Abelian transitive groups such that the topological rank ρ of the infinite Cartesian product of these groups is finite. We consider the corresponding inverse limit of iterated permutational wreath products. By using the geometric language of automorphisms of a certain rooted tree we show that the group is topologically generated by the union of two sets each containing ρ elements and such that both the group generated by S and the group generated by are free Abelian groups of rank ρ. Moreover, the group G generated by the above union decomposes as a semidirect product of the group generated by one of these sets and the normal closure of the other set, and the semigroup generated by this union is a free product of semigroups generated by S and , respectively. We derive other algebraic properties of the group G. In particular, the condition for the groups to be finitely generated as profinite groups and some nontrivial results concerning topological ranks of their subgroups are given in this way. For example, for every our construction gives an explicit and naturally defined finitely generated subgroup of an inverse limit of iterated wreath products of finite Abelian groups such that the rank of this subgroup and the topological rank of its topological closure differ by n.

Abelian subgroups generated by Dehn twists in homeomorphism group

The subgroup of homeoniorphisni group generated by Dehn twists along the set of simple closed pairwise non-homotopic curves with some conditions is studied. It is proved that this group is isomorphic to a free Abelian group of rank k, where k is the number of curves in the set. In the case of an orientable surface, this result is classical.

Totally unimodular nets

p-Periodic nets can be derived from a voltage graph G with voltages in Z(p), the free abelian group of rank p, if the cyclomatic number γ of G is larger than p. Equivalently, one may describe a net by providing a set of (γ - p) cycle vectors of G forming a basis of the subspace of the cycle space of G with zero net voltage. Let M be the matrix of this basis expressed in the edge basis of the 1-chain space of G. A net is called totally unimodular whenever every sub-determinant of M belongs to the set {-1, 0, 1}. Only a finite set of totally unimodular nets can be derived from some finite graph. It is shown that totally unimodular nets are stable under the operation of edge-lattice deletion in a sense that makes them comparable to minimal nets. An algorithm for the complete determination of totally unimodular nets derived from some finite graph is presented. As an application, the full list of totally unimodular nets derived from graphs of cyclomatic numbers 3 and 4, without bridges, is given. It is shown that many totally unimodular nets frequently occur in crystal structures.

Automorphism groups of in nite meta-(locally cyclic) groups

As a continuation of previous work,in this paper,we mainly study the infinite meta-(locally cyclic) groups and their automorphism groups. Let A and B be torsion free abelian groups of rank 1,G be a cyclic group of order n,and let E be extensions of A by G,G by A or A by B respectively. We study the group structure of E as well as their automorphisms,especially their automorphism groups will be given.