Infinite Cyclic Subgroup Research Articles

Monoid presentations of groups by finite special string-rewriting systems

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

On the geometry of infinite cyclic subgroups

In this paper we construct an example of a word metric on an infinite cyclic subgroup. This example shows that subexponential distortion does not obstruct non-trivial growth of connected radii. This answers a question of Gromov [6]. The constructed metric has other pathological properies. Specifically, its asymptotic cone depends on the choice of ultrafilter and scaling sequence.

On the Exponent of the NK0-Groups of Virtually Infinite Cyclic Groups

AbstractIt is known that the K-theory of a large class of groups can be computed from the K-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the K-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of NK0-groups that appear in the calculation of the K0-groups of virtually infinite cyclic groups.

Examples of amalgamated products

The rank of a groupG is the minimal number of elements that generateG. For any natural numbern we construct two groups,G1 of rankr(G1)=n andG2 of rankr(G2)=2n such that their amalgamated product over an infinite cyclic subgroup, malnormal in both factors, is generated by 2n=r(G1)+r(G2)−n elements. We also consider an example of an amalgamated product ofn factors:\(G = \mathop * \limits_{i = 1}^n {}_AG_i \) such thatr(G)=n +1, andr(A)≥1. This example realizes the lower bound given by Weidmann [W1] (see Theorem 2 in the present paper).

Splittings of finitely generated groups over two-ended subgroups

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated groups which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.

On the product of an abelian group and a nilpotent group

We study the structure of the product of an Abelian group and a nilpotent group. Conditions for the existence of a normal subgroup in one of the factors are given. These conditions generalize the known results on the product of two Abelian groups. The statements obtained are used to describe the structure of a product of an infinite cyclic subgroup and a periodic nilpotent subgroup.

The fundamental group at infinity

Let G be a finitely presented infinite group which is semistable at infinity, let X be a finite complex whose fundamental group is G, and let ω be a base ray in the universal covering space X ̃ . The fundamental group at ∞ of G is the topological group π 1 e( X, ω) ≡ lim {π 1 ( X − L)∣L ⊂ X is compact} . We prove the following analogue of Hopf's theorem on ends: π 1 e( X ̃ ,ω) is trivial, or is infinite cyclic, or is freely generated by a non-discrete pointed compact metric space; or else the natural representation of G in the outer automorphisms of π 1 e( X ̃ ,ω) has torsion kernel . A related manifold result is: Let G be torsion free (not necessarily finitely presented) and act as covering transformations on a connected manifold M so that the quotient of M by any infinite cyclic subgroup is non-compact; if M is semistable at ∞ then the natural representation of G in the mapping class group of M is faithful. The latter theorem has applications in 3-manifold topology.

Infinite cyclic verbal subgroups of relatively free groups

We prove that there exist a relatively free group H H and a word w ( x , y ) w(x,y) in two variables such that the verbal subgroup of H H defined by w ( x , y ) w(x,y) is an infinite cyclic group whereas w ( x , y ) w(x,y) has only one nontrivial value in H H .

Hyperbolic elements in negatively curved groups

We show that in a negatively curved groupG the conjugacy class of any infinite cyclic subgroup contains a straight element, an elementg with |g n |=n|g|, and thus the translation number of an element in a negatively curved group is rational with uniformly bounded denominator. We also find an upper bound on the cardinality of a finite normal subgroup.

The Residual Finiteness of Polygonal Products—Two Counterexamples

AbstractWe show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

Quasiconvex subgroups of negatively curved groups

If H is a finitely generated group, then Γ〈 h 1,…, h n 〉, the Cayley graph of H with respect to a finite generating set { h 1,…, h n }, has as vertices the elements of H. There is an edge between the vertices v and w of Γ if vh i = w for some i ∈ {1,…, n}. Theorem. Let H be a negatively curved group. If A is an infinite quasiconvex subgroup of H (i.e. there is a real number ε such that every geodesic in Γ between points of A is within ε of A) then: (1) A has finite index in the normalizer of A in H. (2) If h ε H and hAh -1 is a subset of A then hAh -1 = A. (3) If N is an infinite normal subgroup of H and N ⊂ A, then A has finite index in H. In a negatively curved group, infinite cyclic subgroups are quasiconvex. Hence (1) generalizes a theorem of Gromov's that centralizers of elements of infinite order in a negatively curved group are virtually cyclic. Finitely generated free groups are negatively curved and any finitely generated subgroup of a finitely generated free group is quasiconvex. The special case of (3) where H is free and A is finitely generated is well known.

Infinite Cyclic Normal Subgroups of Fundamental Groups of Noncompact 3- Manifolds

Infinite Cyclic Normal Subgroups of Fundamental Groups of Noncompact 3- Manifolds

Infinite cyclic normal subgroups of fundamental groups of noncompact 3-manifolds

It is shown that an end-irreducible 3 3 -manifold each of whose boundary components is compact and whose fundamental group contains an infinite cyclic normal subgroup is Seifert fibered.

On 4-manifolds homotopy equivalent to circle bundles over 3-manifolds

We give criteria for a closed 4-manifold to be homotopy equivalent to the total space of an S1-bundle over a closed 3-manifold. In the aspherical case the conditions are that the Euler characteristic be 0 and that the fundamental group have an infinite cyclic normal subgroup such that the quotient group has one end and finite cohomological dimension. Under further assumptions on this quotient group we characterize the total spaces of such bundles over $$\mathop {SL}\limits^ \sim $$ -or H2 × E1-manifolds and over E3-, Nil3- or Sol3-manifolds up to s-cobordism and homeomorphism respectively.

On the structure of a real crossed group algebra

We prove that the crossed group algebra A of the infinite dihedral group over the real field defined by the generators a and b, relations b−1 ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ is a principal left ideal ring. This corrects a result of Buzási and provides the missing step towards the classification of finitely generated torsion-free RG-modules for groups G which contain an infinite cyclic subgroup of finite index.

On the structure of a real crossed group algebra

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebraAof the infinite dihedral group ≤a, b|b−1ab=a−1,b2= 1 ≥ over the field R of real numbers: namely in theAdefined byb−lab = a−1, b2= −1, and λa=aλ, λb=bλ for all real λ.The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groupsGwhich have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the presentA. The point of the main result is that new ideas will be needed for understandingA-modules.A number of subsidiary results are concerned with convenient generating sets for left ideals inA.

On the ends of higher dimensional knot groups

An n-knot (S” in Sn+2) is said to be quasi-aspherical if Hn + i(c) = 0 where c is the universal cover of the complement of the knot. S.J. Lomonaco conjectured that all 2-knots are quasi-aspherical. We give an example of a 2-knot whose group has an infinite number of ends and we show that it is a counterexample to Lomonaco’s conjecture. We also prove: (1) every knot whose group has one or two ends is quasi-aspherical; (2) the group of every fibered knot has one or two ends; (3) the class of knots each of whose groups has one or two ends is closed under composition. The theory of ends of finitely generated groups was developed by Heinz Hopf [B] and Hans Freudenthal[6] and is based on Freudenthal’s earlier theory of the ends of a space [5]. All groups in the paper will be finitely generated unless otherwise stated. Excellent references for the theory of the ends of a group are [3, 4, 181. In [B] Hopf proved the following: (1) a group has either 0, 1, 2 or an infinite number of ends, (2) a group has 0 ends if and only if it is finite, (3) a group has two ends if and only if it has an infinite cyclic subgroup of finite index. In [20] C.T.C. Wall sharpened Hopf’s characterization of groups with 2 ends by showing that a group G has 2 ends if and only if it is one of the following two types:

A remark on the integral cohomology of BΓ q

In this paper we provide a negative answer to the question raised in [l] as to whether these results hold over the integers. We do this by constructing enough examples of foliations whose integral pontrjagin rings are non-zero mod p in high dimensions for all odd primes p. These foliations are the “ horizontal ” foliations of bundles with discrete structural groups, Translating this result to a statement about the map v we show that any infinite cyclic subgroup G c Hk(BO,; 2) injects under v *, But if k > 2q Theorem 1 implies that the natural map p: Hk(BTq; 2) + Hk(BIYq; Q) sends v*(G) to zero. Finally we show that whenever the above situation holds the homology of the second space must be infinitely generated in dimension k - 1. This gives our main theorem.

On the group property recognition problem

The unsolvability of the problem of deciding whether a class of finitely presented groups in a (p+3)-letter alphabet has Markov group properties is proved (p is the number of generators of the finitely presented group having a particular property of the kind in question). The problem of deciding whether a class of finitely presented groups in the minimal (p+l)-letter alphabet has Markov properties such that a group having those properties contains an infinite cyclic subgroup is proved to be unsolvable.