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Topics in Combinatorial Group Theory Topics in Combinatorial Group Theory Preface I gave a course on Combinatorial Group Theory at ETH, Zurich, in the Winter term of 1987/88. The notes of that course have been reproduced here, essentially without change. I have made no attempt to improve on those notes, nor have I made any real attempt to provide a complete list of references. I have, however, included some general references which should make it possible for the interested reader to obtain easy access to any one of the topics treated here. In notes of this kind, it may happen that an idea or a theorem that is due to someone other than the author, has inadvertently been improperly acknowledged, if at all. If indeed that is the case here, I trust that I will be forgiven in view of the informal nature of these notes. Acknowledgements I would like to thank R. Suter for taking notes of the course and for his many comments and corrections and M. Schunemann for a superb job of “TEX-ing” the manuscript. I would also like to acknowledge the help and insightful comments of Urs Stammbach. In addition, I would like to take this opportunity to express my thanks and appreciation to him and his wife Irene, for their friendship and their hospitality over many years, and to him, in particular, for all of the work that he has done on my behalf, making it possible for me to spend so many pleasurable months in Zurich. Combinatorial Group Theory iii CONTENTS Chapter I History 1. Introduction ..................................................................1 2. The beginnings ...............................................................1 3. Finitely presented groups ......................................................3 4. More history..................................................................5 5. Higman’s marvellous theorem..................................................9 6. Varieties of groups ...........................................................11 7. Small Cancellation Theory....................................................16 Chapter II The Weak Burnside Problem 1. Introduction .................................................................20 2. The Grigorchuk-Gupta-Sidki groups ...........................................22 3. An application to associative algebras .........................................33 Chapter III Free groups, the calculus of presentations and the method of Reide- meister and Schreier 1. Frobenius’ representation.....................................................35 Combinatorial Group Theory iv 2. Semidirect products..........................................................40 3. Subgroups of free groups are free .............................................45 4. The calculus of presentations .................................................57 5. The calculus of presentations (continued) .....................................62 6. The Reidemeister-Schreier method ............................................70 7. Generalized free products.....................................................73 Chapter IV Recursively presented groups, word problems and some applications of the Reidemeister-Schreier method 1. Recursively presented groups .................................................76 2. Some word problems .........................................................79 3. Groups with free subgroups...................................................80 Chapter V Ane algebraic sets and the representative theory of nitely generated groups 1. Background .................................................................93 2. Some basic algebraic geometry ...............................................94 3. More basic algebraic geometry................................................99 4. Useful notions from topology ................................................101 5. Morphisms .................................................................105 6. Dimension .................................................................112 7. Representations of the free group of rank two in SL(2,C) .....................116 8. Ane algebraic sets of characters............................................122 Chapter VI Generalized free products and HNN extensions 1. Applications................................................................128 2. Back to basics..............................................................132 3. More applicatons ...........................................................137 4. Some word, conjugacy and isomorphism problems ............................147 Chapter VII Groups acting on trees 1. Basic denitions ............................................................151 Combinatorial Group Theory v 2. Covering space theory ......................................................159 3. Graphs of groups ...........................................................161 4. Trees ......................................................................164 5. The fundamental group of a graph of groups .................................167 6. The fundamental group of a graph of groups (continued) .....................169 7. Group actions and graphs of groups..........................................174 8. Universal covers ............................................................180 9. The proof of Theorem 2 ....................................................184 10. Some consequences of Theorem 2 and 3 .....................................185 11. The tree of SL2 ............................................................189 CHAPTER I History 1. Introduction This course will be devoted to a number of topics in combinatorial group theory. I want to begin with a short historical account of the subject itself. This account, besides being of interest in its own right, will help to explain what the subject is all about. 2. The Beginnings Combinatorial group theory is a loosely dened subject, with close connections to topology and logic. Its origins can be traced back to the middle of the 19th century. With surprising frequency problems in a wide variety of disciplines, including dierential equations, automorphic functions and geometry, were distilled into explicit questions about groups. The groups involved took many forms – matrix groups, groups preserving e.g. quadratic forms, isometry groups and Combinatorial Group Theory: Chapter I 2 numerous others. The introduction of the fundamental group by Poincare in 1895, the discovery of knot groups by Wirtinger in 1905 and the proof by Tietze in 1908 that the fundamental group of a compact nite dimensional arcwise connected manifold is nitely presented served to underline the importance of nitely presented groups. Just a short time earlier, in 1902, Burnside posed his now celebrated problem. Problem 1 Suppose that the group G is nitely generated and that for a xed positive integer n xn =1forall x ∈ G. Is G nite? Thus Burnside raised for the rst time the idea of a niteness condition on a group. Then, in a series of extraordinarily inuential papers between 1910 and 1914, Max Dehn proposed and partly solved a number of problems about nitely presented groups, thereby heralding in the birth of a new subject, combinatorial group theory. Thus the subject came endowed and encumbered by many of the problems that had stimulated its birth. The problems were generally concerned with various classes of groups and were of the following kind: Are all the groups in a given class nite (e.g., the Burnside Problem)? Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether or not they are isomorphic? And so on. The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. However much has been accomplished and a wide variety of techniques and methods has been developed with wide application and potential. Some of these techniques have even found a wider use e.g. in the study of so-called free rings and their relations, in generalisations of commutative ring theory, in logic, in topology and in the theory of computing. The reader might wish to consult the book by Chandler, Bruce and Wilhelm Magnus, The History of Combinatorial Group Theory: A Case Study in the History of Ideas, Studies in the History of Mathermatics and the Physical Sciences 9 (1982), Springer-Verlag, New York, Heidelberg, Berlin. Combinatorial Group Theory: Chapter I 3 I do not want to stop my historical account at this point. However, in order to make it intelligible also to those who are not altogether familiar with some of the terms and notation I will invoke, as well as some of the theorems and denitions I will later take for granted, I want to continue my discussion interspersing it with ingredients that I will call to mind as I need them. 3. Finitely presented groups Let G be a group. We express the fact that H is a subgroup of G by writing H G;ifH is a normal subgroup of G we write H/G. Let X G. Then the subgroup of G generated by X is denoted by gp(X). Thus, by denition, gp(X) is the least subgroup of G containing X. It follows that ε1 εn gp(X)={x1 ...xn | xi ∈ X, εi = 1}. We call the product ε1 εn x1 ...xn (xi ∈ X, εi = 1) an X-product. An X-product is termed reduced if xi = xi+1 implies εi + εi+1 =06 (i =1,...,n 1). If G =gp(X) and every non-empty reduced X-product
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