Gilbert Baumslag

Algebraic Geometry over Groups

Classical commutative algebra provides the underpinnings of classical algebraic geometry. In this paper we will describe, without any proofs, a theory for groups that parallels this commutative algebra and that, in like fashion, is the basis of what we term algebraic geometry over groups.

Automatic groups and amalgams

Abstract The objectives of this paper are twofold. The first is to provide a self-contained introduction to the theory of automatic and asynchronously automatic groups, which were invented a few years ago by J.W. Cannon, D.B.A. Epstein, D.F. Holt, M.S. Paterson and W.P. Thurston. The second objective is to prove a number of new results about the construction of new automatic and asynchronously automatic groups from old ones by means of amalgamated products.

The topology of discrete groups

In this paper we explore and exploit a relation between group theory and topology whose existence is suggested by a recent result of D. Kan and W. Thurston [14]: they show that for any connected CW-complex X, there exist a group TX and a map r, :K(TX, 1)+X which induces a homomorphism of TX = rlK(TX, 1) onto riX and gives rise to isomorphisms of homology with respect to all local coefficient systems on X. We amplify this result by showing that the category of pointed, connected CW-complexes and homotopy classes of maps is equivalent to a category of fractions of the category of pairs (G, P), where G is a group and P is a perfect normal subgroup. In other words, homotopy theory can be reconstructed within group theory. In view of this close connection, it seems appropriate to use methods of a geometric character in studying groups. We do this, in particular, for groups corresponding to finite CW-complexes, which we shall call geometrically finite groups; these are the groups for which X(G, 1) has the homotopy type of a finite complex. This implies that G has a finite presentation but is in fact a good deal stronger. For example, such a group must be torsion free. Central for these constructions, as well as those of Kan and Thurston, is the fact that any group can be imbedded in a larger group which has the integral homology of a point; this can be thought of as being analogous to the cone over a topological space. Thus for example a “suspension” of a group can be constructed by amalgamating two cones along the original group. These “acyclic” groups evidently form an important set of building blocks and we investigate both their construction and properties. We show, for example, that geometrically finite groups can be imbedded in geometrically finite acyclic groups, finitely generated groups in finitely generated acyclic groups, and locally finite groups in locally finite acyclic groups. Among the applications of our methods is the result that algebraically closed groups are acyclic. We construct, finally, a functor L of simplicial complexes sharing the properties adduced above for the functor of Kan and Thurston, but with the added property of being effective; i.e., of providing presentations for the groups LX and of taking finite

Generalized triangle groups

A group G is called a triangle group if it can be presented in the form It is well-known that G is isomorphic to a subgroup of PSL 2 (ℂ), that a is of order l, b is of order m and ab is of order n . If then G contains the fundamental group of a positive genus orientable surface as a subgroup of finite index whenever s ( G ) ≤ 1; in particular G is infinite. Furthermore, if s ( G ) G contains a free group of rank 2.

Groups with the same lower central sequence as a relatively free group. II. Properties

0.0. This paper is a continuation of [3], which contains the motivation for this work. Many of the terms and notations used in [3] will be used throughout this paper without any explicit reference. Suppose that 3 is a variety of groups. A group P is termed parafree(2) in 3, or a parafree 3-group, or simply parafree if there is no question as to which variety is involved, if (i) P is a 3-group i.e., P ce 3; (ii) P is residually nilpotent; (iii) P has the same lower central sequence(2) as some free 3-group. The object of this paper is to investigate the properties of such parafree groups with the basic hope (which is amply fulfilled) that many properties of the relevant free group persist in the corresponding parafree group. 0.1. The first section is preliminary in nature being essentially elementary. First we prove that if P is a parafree 3-group of finite rank and if N is a normal subgroup of P whose quotient P/N is again parafree in 3 of the same rank as P, then N= 1. This almost obvious fact implies that parafree groups of finite rank are hopfian. Of course parafree groups of rank one are cyclic. It follows easily from this that a parafree group of rank two is freely decomposable if and only if it is free. Next we consider absolutely parafree groups i.e., groups parafree in the variety of all groups. For such groups we prove that free products and free factors are again parafree. 0.2. In ?2 we complement the results in [3] by obtaining some more, rather different, parafree groups. We first consider one-relator groups.

The algorithmic theory of polycyclic-by-finite groups☆

A group is said to be polycyclic-by-finite, or a PF-group for short, if it has a polycyclic normal subgroup of finite index. Equivalently, PF-groups are exactly the groups which have a series of finite length whose infinite factors are cyclic. By a well-known theorem of P. Hall every PF-group is finitely presented-and in fact PF-groups form the largest known sectionclosed class of finitely presented groups. It is this fact that makes PF-groups natural objects of study from the algorithmic standpoint. The general aim of the algorithmic theory of PF-groups can be described as the collection of information about a PF-group which can, in principle

On finitely presented metabelian groups

My objective here today is to describe what is known about the defining relations of finitely generated metabelian groups. This is not a difficult task because very little is known and much of what is known is of very recent origin. Despite this meagre knowledge, the results obtained so far suggest that the theory of finitely presented solvable groups is far richer than one might have suspected. It is for this reason that I have chosen to discuss finitely presented metabelian groups at this time.

Some Problems on One-Relator Groups

A baler having an improved pickup counterbalance mechanism is disclosed. The improved mechanism includes biased pivot cranks connected to each end of the pickup and a variable length connector slideably fixed between the biasing means to provide a single adjustment for effectively equalizing the counterbalance forces on the two lateral ends of the pickup.

Computable algebra and group embeddings

This work grew out of our investigations of the infinitely generated subgroups of finitely presented groups. A useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in [l]. This technique was then applied in [ 1, 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group. It became apparent from [2] that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups. In this paper these extensions are carried out with the consequent application that every countable locally abelian-by-nilpotent group can be embedded in a finitely presented group. Much of the effective commutative algebra we require seems to be known. (See for instance [6, 9, 14, 171.) Unfortunately a systematic account is not available. In particular we need a proof of the Hilbert basis theorem which

Discriminating Completions of Hyperbolic Groups

A group G is called an A-group, where A is a given Abelian group, if it comes equipped with an action of A on G which mimics the way in which Z acts on any group. This action is codified in terms of certain axioms, all but one of which were introduced some years ago by R. C. Lyndon. For every such G and A there exists an A-exponential group GA which is the A-completion of G. We prove here that if G is a torsion-free hyperbolic group and if A is a torsion-free Abelian group, then the Lyndons type completion GA of G is G-discriminated by G. This implies various model-theoretic and algorithmic results about GA.