Abstract

Bounded languages are a class of formal languages which includes all context free languages of polynomial growth. We prove that if a finitely generated group G admits a combing by a bounded language and this combing satisfies the asynchronous fellow traveller property, then either G is virtually abelian, or else G contains an element g of infinite order such that gn and gm are conjugate for some 0 < n < m.

The introduction of automatic groups [5] has precipitated a host of questions about the roles which formal language theory and geometry play in the study of normal forms for finitely generated groups, particularly groups which arise in geometric settings. For example, when a group G is given as the fundamental group of a compact Riemannian manifold, words in a fixed set of generators for G have a natural interpretation as paths in the universal cover of the manifold; it is natural to ask how the geometry of the manifold is reflected in the linguistic complexity of normal forms for elements of G. The results presented here and in [3] can be interpreted as providing a partial answer to this question in the case where the manifold under consideration is a quotient of a nilpotent Lie group.

Introduction

One possibility to show that a group G with finite semigroup generating system S has rational growth series is to exhibit a regular language L ⊆ S* consisting of geodesic words and mapped bijectively onto G. Machi and Schupp have even conjectured that the existence of such an L is equivalent to the rationality of the growth series ([2], conjecture 8.7).

The aim of this paper is to show that this approach does not work for nilpotent groups. More precisely, we show that if G is 2-step nilpotent with maximal free abelian quotient G and S is any finite set of semigroup generators for G, then for every regular and geodesic (with respect to G) language L ⊆ S*, the natural map L → G has finite fibers. We conjecture that this holds for all nilpotent groups. If G is not virtually abelian, this implies in particular that L cannot be mapped bijectively onto G.

This also gives a counterexample to the conjecture of Machi and Schupp, for it is known that the discrete Heisenberg group with its standard generating set,

has rational growth series [3], but our theorem implies that no regular and geodesic language can be mapped bijectively onto H.

Notations and Definitions

Let G denote an arbitrary finitely generated group.

If P is a group of operators on a group A, then we denote by dp(A) the minimal number of generators of A as a P-group.

Suppose G is a group and F/N is a presentation of G. Then F acts on N by conjugation and induces an action of G on Nab. This ℤG-module is called the relation module of the presentation F/N.

Definition. A presentation F/N of a group G is said to have a relation gap if dG(Nab) is strictly less than dF(N).

It is an open problem whether there exists a presentation that has a relation gap (see Harlander [H1], [H2] and Baik, Pride [B-P]). Such a presentation would be interesting not only to group theorists. In [D] Dyer shows that a presentation with a relation gap could be used to settle an open question concerning complexes dominated by a 2-complex (see also Wall [W] and Ratcliffe [R]).

In [H1] and [H2] the author studies groups that have cyclic relation modules. Such groups are quotients G/P, with G a one-relator group, say presented by 〈X ∣ r〉, and P a perfect normal subgroup G. Now if P is not of the form 〈w〉F / 〈r〉F, where 〈w〉F denotes the normal closure of the element w of the free group F on X, then G/P has a presentation with a relation gap.

1. The idea of a Λ-tree, where Λ is an ordered abelian group, was introduced in [9]. We shall reproduce the definition shortly, but for an account of the basic theory of Λ-trees we refer to [1]. In the special case Λ = ℤ, ℤ-trees are closely related to simplicial trees (trees in the ordinary graph-theoretic sense). The connection is spelt out in Lemma 4 below, which shows that Λ-trees may be viewed as generalisations of simplicial trees. However, there are other notions of generalised tree in the literature, and our purpose here is to consider two of these, and their relation to Λ-trees.

Firstly there is what we call an order tree. This is a partially ordered set {P, ≤) such that the set of predecessors of any element is linearly ordered, that is, for all x, y, z ∈ P, if x ≤ z and y ≤ z, then either x ≤ y or y ≤ x. It is also convenient to assume that P has a least element (this can always be arranged just by adding one). By choosing a point in a Λ-tree, it is possible to make the Λ-tree into an order tree. We shall show that, conversely, any order tree (P, ≤) can be embedded in a Λ-tree for some suitable Λ, so that the ordering on P is induced from the ordering on the Λ-tree defined by the (image of) the least element of P.

Abstract

We outline a proof that if G is a soluble or linear group of type (FP)∞ then G has finite virtual cohomological dimension. The proof depends on hierarchical decompositions of soluble and linear groups and also makes use of a recently discovered generalized Tate cohomology theory. A survey of this complete cohomology is included. The paper concludes with a review of some open problems.

Preface

The first part of this article is based on a lecture delivered at the conference. It concerns the proof that soluble and linear groups of type (FP)∞ have finite vcd. More general results have been published in [21], but in order to make the key new arguments widely accessible I thought it worthwhile going through the special cases again. Several technical problems can be avoided this way, and I hope that this will make for clarity.

At the conference, a number of people asked about the generalized Tate cohomology theory which plays such a crucial and somewhat miraculous role. For this reason I have included a detailed account in §4.

In the last section, some problems and questions are discussed which I did not have time to cover in the lecture. Some of the results in this section have not been published elsewhere.

Introduction

Let G be a group. This paper studies projective resolutions P* ↠ ℤ of the trivial module ℤ over the group ring ℤG.

AUTHOR'S NOTE. This paper was written in 1980, but was not published at that time. The reference made to it in Miller's survey article [10], however, made me think that it might be worth publishing. It is unchanged except for the deletion of some remarks that either are outdated or no longer seem interesting. I am grateful to Professor Miller for his interest in the paper.

Introduction.

In the first part of this paper we give proofs of two embedding theorems for groups, and a version of the Adjan-Rabin construction for showing that many group-theoretic decision problems are unsolvable, which seem to be simpler than the standard ones. In the second part we consider some specific undecidability questions.

In [5], Higman, Neumann and Neumann proved the following theorem (see also [11]).

Theorem 1 Every countable group G can be embedded in a 2-generator group H. If G is n-relator, then H may be taken to be n-relator.

P. Hall (unpublished) proved that every countable group can be embedded in a finitely generated simple group. This was sharpened by Goryushkin [3] and Schupp [13] to

Theorem 2 Every countable group can be embedded in a 2-generator simple group.

This final chapter has two purposes. A few topics not considered earlier are discussed briefly; usually there is a central problem which has served as a focus for research. Then there is a list of assorted problems in other areas, and some recommended reading for further investigation of some of the main subdivisions of combinatorics.

Computational complexity

This topic belongs to theoretical computer science; but many of the problems of greatest importance are of a combinatorial nature. In the first half of this century, it was realised that some well-posed problems cannot be solved by any mechanical procedure. Subsequently, interest turned to those which may be solvable in principle, but for which a solution may be difficult in practice, because of the length of time or amount of resources required. To discuss this, we want a measure of how hard, computationally, it is to solve a problem. The main difficulty here lies in defining the terms!

Problems .

Problems we may want to solve are of many kinds: anything from factorising a large number to solving a system of differential equations to predict tomorrow's weather. In practice, we usually have one specific problem to solve; but, in order to do mathematics, we must consider a class of problems.