Free Group Rings Narain Gupta

AMERICAn MATHEMATICAL SOCIETY VOLUME 66

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Free Group Rings

COnTEMPORARY MATHEMATICS

Volume&&

Free Group Rings

Narain Gupta1 Editor

AMERICAn MATHEMATICAL SOCIETY Providence • RIIJda Island

Irwin Kra, EDITORIAL BOARD

W. H. Jaco managing editor

Thomas F. Banchoff M. Salah Baouendi

Gerald J. Janusz Jan Mycielski Johannes C. C. Nitsche

Alan D. Weinstein

1980 Mathematics Subject Classification (1985 Revision). Primary 20F05, 20F10, 20F12, 20F14, 20F26, 20C05, 20C07, 16A27, 20H25.

Library of Congress Cataloging-In-Publication Data Gupta, Narain.

Free group rings.

(Contemporary mathematics, ISSN 0271-4132; v. 66) Bibliography: p. 1. Group rings. I. Title. II. Series: Contemporary mathematics (American Mathematical

Society); v. 66.

QA171.G88 1987 5121.4 87-12427 ISBN 0-8218-5072-5 (alk. paper)

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Table of Contents

Preface

CHAPTER I Magnus Embeddings and Free Differential Calculus

1. Some faithful 2 X 2 matrix representations 2. Free differential calculus 3. The fundamental theorem of free group rings 4. The Poincare-Birkhoff-Witt Theorem

CHAPTER II Applications of Magnus Embedding 1. Some properties of F/R' 2. Some algorithmic problems in F/R' 3. Faithfulness of relation modules 4. Residual nilpotence of F/R'

CHAPTER III Fox Subgroups of Free Groups 1. A faithful matrix representation of F/F(n,R) 2. Identification of F(2,R) 3. Identification of F(n,R) (F/R finite) 4. A filtration of the lower central series of F 5. A solution of the Fox problem 6. Structure of Fox modules 7. The centre of F/F(n,R), n ~ 2

CHAPTER IV Dimension Subgroups 1. Sjogren's theorem 2. A counter-example due to rips 3. Some preliminary results on dimension subgroups 4. Dimension subgroups of metabelian groups 5. Structure of the fourth dimension subgroup 6. Dimension subgroups of metabelian p-groups

CHAPTER V Generalized Magnus Embeddings 1. A faithful matrix representation of ?l:F/-H (~ s 2. A faithful matrix representation of 1lF/~ 1 ···~n

vii

ix

1 1 7

11 19

23 23 32 39 43

46 46 48 49 51 53 58 61

64 64 76 80 87 94 98

104

6> 104 107

TABLE OF CONTENTS

3. Residual nilpotence of F/F n (1+~ 1 - •• ~n) 4. Identification of F n (1+~ 1 ••• ~n) for ~all n 5. Some applications of the generalized embeddings 6. The problem section

BIBLIOGRAPHY

viii

109 113

116 118

121

Preface

This MONOGRAPH deals with some aspects of linear techniques in Combinatorial Group Theory which have their origin in the work of Wilhelm Magnus in the thirties. The central theme is the identification and properties of those subgroups of free groups which are induced by certain ideals of the free group rings, the integral group rings of free groups. The subject has been developed extensively in the literature and my intention is to give a systematic and comprehensive account of some of its devel-opments in a contemporary style. The choice of topics presented and their respective treatment is inevitably influenced by my own research over a long period. A brief outline of the topics dealt with is as follows.

CHAPTER I is introductory. Here we begin with a faithful 2 x 2 matrix representation of the quotient ring 2ZF/~p·, where 6 2ZF (F-1) is the augmentation ideal of the free group ring 2ZF and ~ is any ideal of 2ZF contained in 6. l'7ith ~ = ~ = Z':F (R-1), R normal in F., we prove Schumann's result that F n (1+~6) R', the commutator subgroup of R. From this we

::::::

deduce the well-known Magnus embedding of F/R' into a suitable ring of 2 x 2 matrices. After a brief introduction to Fox's Free Differential Calculus we formulate the so-called FOX PROBLEM: Identify the subgroups of F induced by the ideals ~Pn for n :<!: 1. In the convenient language of free groups .rings, we also formulate the well-known DIMENSION SUBGROUP PROBLEM: Identify the subgroups of F induced by the ideals ~+~n for n :<!: 1. We con-clude the chapter with a self-contained new proof of the Funda-mental Theorem of Free Group Rings due to Magnus, namely, the ideal 6n induces the n-th term of the lower central· series of F for each n :<!: 1.

CP~TER II is designed to illustrate the fact that the Magnus embedding provides a powerful tool in group theory. In particular,

ix

X PREFACE

we derive some of the well-known results about F/R' including the solution of the coniugacy problem of F/R' due to Remeslenni-kov & Sokolov and Passi's theorem about the faithfulness of R/R' as a 2Z(F/R)-module.

In CHAPTER III we give a detailed account of the Fox Problem including its recent solutions due to Gupta and Yunus. We also include the structure of the Fox modules and their fixed sub-modules due to Gupta & Passi.

CHAPTER IV contains an up-to-date development of the Dimen-sion Subgroup Problem. Simplifications due to Hartley, Cliff & Hartley and Gupta yield an easy access to Sjogren's Fundamental Theorem from which most of the earlier results are deduced as corollaries. We also include an accessible account of Rips' counter-example to the Dimension Subgroup Conjecture. The remain-der of the chapter is devoted to the dimension subgroups of meta-belian groups. Here we include a theorem of Gupta, Hales & Passi about the dimension subgroups of finitely generated metabelian groups. We also include Tahara's description of the fourth di-mension subgroups and conclude the chapter with a new result about the dimension subgroups of metabelian p-groups.

In CHAPTER V we describe a faithful matrix representation of the quotient groups F/F n (1+11 ···1n)' n ~ 2, 1i = 2ZF(Ri-l), Ri normal in F, due to Lewin and Yunus. We derive some important results about the residual nilpotence of these groups due to Yunus and C.K. Gupta & Passi. we point out some recent developments in the areas of generalized embeddings due to Kuz'min and Stohr, and conclude the chapter with some general open problems in the area of free group rings.

This monograph is aimed at graduate students and other re-searchers in the area of combinatorial group theory. In particu-lar, we assume some familiarity with the general terminology of free groups and group rings. Lack of my own expertise in the areas of homological and geometrical aspects of free group rings has prevented me from introducing these developments into the text. For these topics the reader is referred to the Lecture Notes by Karl Gruenberg (1970) and Joan Birman (1974). For other aspects of Group Rings the reader is referred, in particular, to Passman (1977), Sehgal (1978) and Passi (1979).

PREFACE xi

The idea for writing a monograph on Free Group Rings was originally suggested by Inder Bir Passi in 1980 after I had given a series of lectures at the Centre for Advanced Study in Mathe-matics, Chandigarh. Much of the material presented in Chapters III and IV has been developed since that time. I thank Inder Bir Passi for his help in organizing and collaborating on various parts of the Monograph. I also thank C. Kanta Gupta, Brian Hartley and Roger Lyndon for their encouragement during this project. This work was completed while I held a Killam Research Fellowship from the Canada Council and a research grant from the Natural Sciences and Engineering Research Council of Canada.

Narain Gupta Winnipeg, January 1987