Infinite Cyclic Group Research Articles

A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups

We consider a group presented as $G(p,q) = \langle x, y|x^p = y^q\rangle$, with integers $p$ and $q$ satisfying $2 \leq p \leq q$. The group is an amalgamated free product of two infinite cyclic groups and is geometrically realized as the fundamental group of a Seifert fiber space over the 2-dimensional disk with two cone points whose associated cone angles are $\frac{2\pi}{p}$ and $\frac{2\pi}{q}$. We present a formula for the spherical growth series of the group $G(p,q)$ with respect to the generating set $\{x,y,x^{-1}, y^{-1}\}$, from which a rational function expression for the spherical growth series of $G(p,q)$ is derived concretely, once $p$ and $q$ are given. In fact, an elementary computer program constructed from the formula yields an explicit form of a single rational fraction expression for the spherical growth series of $G(p,q)$. Such expressions for several pairs $(p,q)$ appear in this paper. In 1999, C. P. Gill already provided a similar formula for the same group. The formula given here takes a different form from his formula, because the method we used here is independent of that introduced by him.

On classical upper bounds for slice genera

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

ON A QUESTION OF A. SOZUTOV

In the paper an answer to a problem posed by A.I. Sozutov in the Kourovka Notebook is given. The solution is based on some modification of the method that was proposed for constructing a non-abelian analogue of the additive group of rational numbers, i.e. a group whose center is an infinite cyclic group and any two non-trivial subgroups of which have a non-trivial intersection.

Topological rigidity for FJ by the infinite cyclic group

We call a group FJ if it satisfies the [Formula: see text]- and [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. We show that if [Formula: see text] is FJ, then the simple Borel conjecture (in dimensions [Formula: see text]) holds for every group of the form [Formula: see text]. If in addition [Formula: see text], which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions [Formula: see text]) holds for [Formula: see text]. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group [Formula: see text] satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text], then any semi-direct product [Formula: see text] also satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. Our result is indeed more general and implies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

Growth series of crossed and two-sided crossed products of cyclic groups

Abstract We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

On lengths of Hℤ-localization towers

In this paper, the Hℤ-length of different groups is studied. By definition, this is the length of the Hℤ-localization tower or the length of the transfinite lower central series of Hℤ-localization. It is proved that, for a free noncyclic group, its Hℤ-length is ≥ ω+2. For a large class of ℤ[C]-modules M, where C is an infinite cyclic group, it is proved that the Hℤ-length of the semi-direct product M ⋊ C is ≤ ω + 1 and its Hℤ-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M, such that M⋊C is finitely presented and H2(M ⋊ C) is finite.

On the Centralizer Dimension and Lattice of Generalized Baumslag–Solitar Groups

A generalized Baumslag–Solitar group (a GBS group) is a finitely generated group G acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups. Each GBS group is the fundamental group π1(A) of some labeled graph A. We describe the centralizers of elements and the centralizer lattice. Also, we find the centralizer dimension for GBS groups if A is a labeled tree.

On Bousfield's problem for solvable groups of finite Prüfer rank

For a group G and R=Z,Z/p,Q we denote by GˆR the R-completion of G. We study the map Hn(G,K)→Hn(GˆR,K), where (R,K)=(Z,Z/p),(Z/p,Z/p),(Q,Q). We prove that H2(G,K)→H2(GˆR,K) is an epimorphism for a finitely generated solvable group G of finite Prüfer rank. In particular, Bousfield's HK-localisation of such groups coincides with the K-completion for K=Z/p,Q. Moreover, we prove that Hn(G,K)→Hn(GˆR,K) is an epimorphism for any n if G is a finitely presented group of the form G=M⋊C, where C is the infinite cyclic group and M is a C-module.

On the Lee classes of locally conformally symplectic complex surfaces

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of H 1 dR (S, R), or a single point. In the latter case, we show that S must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham co-homology classes of Lee forms of locally conformally Kahler structures of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of H 1 dR (S, R), a single point or the empty set. We give a characterization of Enoki surfaces in terms of the existence of a special foliation, and obtain a vanishing result for the Lichnerowicz-Novikov cohomology groups on the class VII compact complex surfaces with infinite cyclic fundamental group.

ON THE ASSOCIATED GROUP OF THE FREE PRODUCT OF A QUASI-FREE GROUP AND A FINITE GROUP

Ponte Academic JournalDec 2018, Volume 74, Issue 12 ON THE ASSOCIATED GROUP OF THE FREE PRODUCT OF A QUASI-FREE GROUP AND A FINITE GROUPAuthor(s): MAHMOOD, R. M. S ,MOHAMMAD AL-HAWARIJ. Ponte - Dec 2018 - Volume 74 - Issue 12 doi: 10.21506/j.ponte.2018.12.8 Abstract:A group is termed a quasi-free group if it is a free product of cyclic groups of any order. Free groups being free product of infinite cyclic groups show that quasi-free groups are the free product of a free group and a number of finite cyclic groups. In this paper we show that for any group G there exists a group denoted H(G) and is called the associated group of G and show that if F is a quasi-free group and G is any group then H(F) is trivial and H(FG)  H(G). 2010 Mathematics Subject Classification: Primary 16U60, 20C05, 16S34, 20E06; Secondary 20C10, 20C40 Download full text:Check if you have access through your login credentials or your institution Username Password

On quasi-convex null sequences in infinite cyclic groups

Abstract We prove that for certain irrational numbers α, there exist quasi-convex null sequences in the subgroup generated by α + ℤ {\alpha+\mathbb{Z}} in the torus.

On central extensions of SL(2,F) admitting left-orderings

For an arbitrary euclidean field F we introduce a central extension (G(F),Φ) of SL(2,F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order-preserving bijections of the convex hull of Q in F. If F=R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2,R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.

A family of irretractable square-free solutions of the Yang–Baxter equation

Abstract A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova’s Strong Conjecture [19, Strong Conjecture 2.28 (I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova’s Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.

ON THE TOTAL COMPONENT OF THE PARTIAL SCHUR MULTIPLIER

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

A chord diagram of a ribbon surface-link

A ribbon chord diagram, or simply a chord diagram, of a ribbon surface-link in the 4-space is introduced. Links, virtual links and welded virtual links can be described naturally by chord diagrams with the corresponding moves, respectively. Some moves on chord diagrams are introduced by overseeing these special moves. Then the faithful equivalence on ribbon surface-links is stated in terms of the moves on chord diagrams. This answers questions by Nakanishi and Marumoto affirmatively. The faithful TOP-equivalence on ribbon surface-links derives the same result. By combining a previous result on TOP-triviality of a surface-knot, a ribbon surface-knot is DIFF-trivial if and only if the fundamental group is an infinite cyclic group. This corrects an erroneous proof in Yanagawa's old paper.

Periodicity of self-injective algebras of polynomial growth

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.

Вычисление рангов групп центральных единиц целочисленных групповых колец конечных групп

The study of central units (central invertible elements) of integral group rings is encountered to difficult calculations almost everywhere, both in the case of finding of individual central unit and in the case of describing of group of central elements. By virtue of torsion part of central unit group is trivial (up to sign those are elements of center group) it is more interesting to find data about torsion free part that is direct product infinite cyclic groups. The number of such infinite factors is the rank of central unit group. Therefore the ranks of central unit groups of integral group rings of finite groups are the very important characteristic those groups. So that the computation ranks of central unit groups has big interest for study of central unit groups. In the paper we point out the formulas for computation of ranks in general case and some important particular cases. On the base of those formulas we compute the ranks in quite large ranges. We used computer algebra system GAP. The results are shown on tables and graph.

Braid groups of imprimitive complex reflection groups

We obtain new presentations for the imprimitive complex reflection groups of type (de,e,r) and their braid groups B(de,e,r) for d,r≥2. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group B(de,e,r) is a semidirect product of the braid group of affine type A˜r−1 and an infinite cyclic group. Elements of B(de,e,r) are visualised as geometric braids on r+1 strings whose first string is pure and whose winding number is a multiple of e. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group B(de,e,r) is strongly translation discrete.

On the functional equation x + f (y + f(x)) = y + f (x + f(y)), II

For an abelian group (G, + ,0) we consider the functional equation $$f : G \to G, \,\, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G),\qquad \qquad\quad (1)$$ (1) together with the condition $$f(0) = 0. \qquad\qquad\qquad\qquad\qquad (0)$$ (0) The main question is that of existence of solutions of (1) \({\wedge}\) (0), specifically in the case when G is the direct sum \({\mathbb{Z}_n^{(J)}}\) of copies of a finite or infinite cyclic group (Theorems 3.2 and 4.20).

Taut foliations and the action of the fundamental group on leaf spaces and universal circles

Let $F$ be a leafwise hyperbolic taut foliation of a closed 3-manifold $M$ and let $L$ be the leaf space of the pullback of $F$ to the universal cover of $M$. We show that if $F$ has branching, then the natural action of $\pi_1(M)$ on $L$ is faithful. We also show that if $F$ has a finite branch locus $B$ whose stabilizer acts on $B$ nontrivially, then the stabilizer is an infinite cyclic group generated by an indivisible element of $\pi_1(M)$.