General Group Research Articles

Fibonacci Sequences in Groups

When this sequence is periodic, its fundamental period is called the Fibonacci length of (x1, x2) in G. When G is a finite 2-generator group, the minimum of these lengths over all generating pairs defines an invariant λ(G) of G. After briefly listing some known results, we launch the quest for infinite groups of finite Fibonacci length by giving three modest examples and conclude with a selection of open problems.

A 2-generated just-infinite profinite group which is not positively generated

We construct a 2-generated profinite group which is just-infinite and not positively generated.

ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

Some 3-manifolds with 2-generated fundamental group

We describe a family of graph manifolds that have 2-generated fundamental group but do not admit a Heegaard splitting of genus 2. They are closely related to the examples provided by Boileau and Zieschang if these are considered as (non-proper) graph manifolds.

On capable $p$-groups of nilpotency class two

A group is called capable if it is a central factor group. Let ${\mathcal{P}}$ denote the class of finite $p$-groups of odd order and nilpotency class 2. In this paper we determine the capable 2-generator groups in ${\mathcal{P}}$. Using the explicit knowledge of the nonabelian tensor square of 2-generator groups in ${\mathcal {P}}$, we first determine the epicenter of these groups and then identify those with trivial epicenter, making use of the fact that a group has trivial epicenter if and only if it is capable. A capable group in ${\mathcal{P}}$ has the two generators of highest order in a minimal generating set of equal order. However, this condition is not sufficient for capability in ${\mathcal{P}}$. Furthermore, various homological functors, among them the exterior square, the symmetric square and the Schur multiplier, are determined for the 2-generator groups in ${\mathcal{P}}$.

Subgroups of powerful groups

We consider subgroups of powerfulp-groups. In particular, we give a new proof that allp-groups are sections of powerfulp-groups, give necessary and sufficient conditions for a 2-generator group to be a normal subgroup of a powerfulp-groups, and show thatp-groups of class 2, orp-groups with a cyclic commutator subgroup, are such normal subgroups.

Hyperbolicity of some 2-generated one-relator groups

Abstract We give the description of word hyperbolic groups of the form (a, b, t; t-1at = b, t-1bt = w(a, b)). The research was supported by the Russian Foundation for Basic Research, grant 00-01-00767.

THE 2-GENERATOR RESTRICTED BURNSIDE GROUP OF EXPONENT 7

We report on our construction of a power-commutator presentation for R(2,7), the largest finite 2-generator group of exponent 7. Our calculations show that R(2,7) has order 720416, nilpotency class 28, and derived length 5. The calculations also imply that the associated Lie ring of R(2,7) satisfies relations which are not consequences of the multilinear identities which hold in the associated Lie rings of groups of exponent 7.

2-Generator arithmetic Kleinian groups III

This paper forms part of the program to identify all the 2-generator arithmetic Kleinian groups. Here we identify all conjugacy classes of such groups with one generator parabolic and the other generator elliptic. There are exactly $14$ of these and exactly $5$ Bianchi groups in their commensurability class, namely $\mathrm{PSL}(2,{\mathcal O}_d)$ for $d=1,2,3,7$ and $15$. This complements our earlier identification of the $4$ arithmetic Kleinian groups generated by two parabolic elements.

Group Quasivarieties with Infinitely Many Maximal Subquasivarieties

We construct an example of a 2-generated group G and describe a set of proper maximal subquasivarieties in the quasivariety qG. This set proves to be infinite, which gives an affirmative answer to Question 14.25 posed by A. I. Budkin in the “Kourovka Notebook.” Moreover, it is shown that every proper subquasivariety in qG is contained in a proper maximal one.

CONSTRUCTION OF A REGULARLY EXHAUSTING SEQUENCE FOR GROUPS WITH SUBEXPONENTIAL GROWTH

It is proved that every group with subexponential growth contains a regularly exhausting sequence with polynomial growth. A consequence is answering in the negative Question 14.27 in [6] — namely, a continuum of non-isomorphic 2-generated groups are constructed which each have a regularly exhausting sequence with polynomial growth and which are but not groups with polynomial growth.

Subnormal Embedding Theorems for Groups

In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results ob- tained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal em- beddings of finite groups into finite groups with additional properties we bring some illustration to a result of H. Heineken and J. Lennox.

The Complexity of theA B CProblem

We present a deterministic polynomial-time algorithm for the A B C problem, which is the membership problem for 2-generated commutative linear semigroups over an algebraic number field. We also obtain a polynomial-time algorithm for the (easier) membership problem for 2-generated abelian linear groups. Furthermore, we provide a polynomial-sized encoding for the set of all solutions.

Recalcitrance in groups

Motivated by a well-known conjecture of Andrews and Curtis, we consider the question as to how in a given n-generator group G, a given set of n “annihilators” of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (”elementary M-transformations”) by means of which every annihilating n-tuple of G can be transformed into a generating n-tuple. We show that in the classes of finite and soluble groups, having zero recalcitrance is equivalent to nilpotence, and that a large class of 2-generator soluble groups has recalcitrance at most 3. Some examples and remarks are included.

2-generator arithmetic Kleinian groups

Abstract We show that among the infinitely many conjugacy classes of finite co-volume Kleinian groups generated by two elements of finite order, there are only finitely many which are arithmetic. In particular there are only finitely many arithmetic generalized triangle groups. This latter result generalizes a theorem of Takeuchi.

Computer-based calculations for transmetabelian groups of exponent 16, 25, and 27

A new stage of research on 2-generated transmetabelian groups of primary exponent is announced, which makes it possible to investigate groups of exponent 16, 25,and 27.More complete data describing the current state of studies are available via the Internet at http: //www.pdmi.ras.ru/∼ yuri/transmet. Bibliography: 7titles.

Automorphisms of one-relator groups

It is a well known fact that every group G has a presentation of the form G=F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of the group of automorphisms Aut (G), we consider the subgroup Stab (R) ⊆ Aut (F) of those automorphisms of F that stabilize R and ask whether the natural homomorphism Stab (R) → Aut (G) is onto; if it is, we can try to determine its kernel.Both parts of this task are usually quite hard. The former part received considerable attention in the past, whereas the more difficult part (determining the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condition. Then, we address a somewhat easier case of 2-generator (not necessarily one-relator) groups and determine the kernel of the above-mentioned homomorphism for a rather general class of those groups.

Groups with exponent six

Burnside asked questions about periodic groups in his influential paper of 1902. The study of groups with exponent six is a special case of the study of the Burnside questions on which there has been significant progress. It has contributed a number of worthwhile aspects to the theory of groups and in particular to computation related to groups. Finitely generated groups with exponent six are finite. We investigate the nature of relations required to provide proofs of finiteness for some groups with exponent six. We give upper and lower bounds for the number of sixth powers needed to define the largest 2-generator group with exponent six. We solve related questions about other groups with exponent six using substantial computations which we explain.

Some p-groups of Weilandt length three

Finite p-groups of Wielandt length 1 are groups in which every subgroup is normal and are Dedekind groups. When the prime is odd therefore a finite p-group of Wielandt length 1 is Abelian. For an odd prime, a finite p-group of Wielandt length 2 has nilpotency class at most 3 and for such a goup to have class 3 there must be a 2-generator subgroup of this class. In this paper it is shown that for any prime p > 3 a finite p-group of Wielandt length 3 has nilpotency class at most 4, and for such a group to have class 4 there must be a 2-generator subgroup with this class. Two families of p-groups of Wielandt length 3 are described. One is a family of 3-generator groups with the property that each group modulo its Wielandt subgroup has class 2, the other is a family of 2-generator groups with the property that each group modulo its Wielandt subgroup has class 3.

Continuously many quasiisometry classes of 2-generator groups

We construct continuously many quasiisometry classes of torsion-free 2-generator small cancellation groups.