Roger C. Lyndon

Bio: Roger C. Lyndon is an academic researcher from University of Michigan. The author has contributed to research in topics: Free group & Group (mathematics). The author has an hindex of 14, co-authored 29 publications receiving 5054 citations.

Combinatorial Group Theory

Abstract: Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7. Quadratic sets of Word 8. Equations in Free Groups 9. Abstract Lenght Functions 10. Representations of Free Groups 11. Free Pruducts with Amalgamation Chapter II. Generators and Relations 1. Introduction 2. Finite Presentations 3. Fox Calculus, Relation Matrices, Connections with Cohomology 4. The Reidemeister-Schreier Method 5. Groups with a Single Defining Relator 6. Magnus' Treatment of One-Relator Groups Chapter III. Geometric Methods 1. Introduction 2. Complexes 3. Covering Maps 4. Cayley Complexes 5. Planar Caley Complexes 6. F-Groups Continued 7. Fuchsian Complexes 8. Planar Groups with Reflections 9. Singular Subcomplexes 10. Sherical Diagrams 11. Aspherical Groups 12. Coset Diagrams and Permutation Representations 13. Behr Graphs Chpter IV. Free Products and HNN Extensions 1. Free Products 2. Higman-Neumann-Neumann Extensions and Free Products with Amalgamation 3. Some Embedding Theorems 4. Some Decision Problems 5. One-Relator Groups 6. Bipolar Structures 7. The Higman Embedding Theorem 8. Algebraically Closed Groups Chapter V. Small Cancellation Theory 1. Diagrams 2. The Small Cancellation Hypotheses 3. The Basic Formulas 4. Dehn's Algorithm and Greendlinger's Lemma 5. The Conjugacy Problem 6. The Word Problem 7. The Cunjugacy Problme 8. Applications to Knot Groups 9. The Theory over Free Products 10. Small Cancellation Products 11. Small Cancellation Theory over free Products with Amalgamation and HNN Extensions Bibliography Index of Names Subject Index

Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series

Abstract: The quotient groups Qn(G) =GnGn+i of the lower central series G = G1 D G, D G, D * * of a finitely generated group G are finitely generated abelian groups. Our object is to develop an algorithm for the calculation of Qn from any given finite presentation of G. As a preliminary step, the special case of a free group X is considered. It is known [2], [7] that, for a free group X of rank q, the group Qn(X) is a free abelian group whose rank is the Witt number 0b,(q), and a basis for QJ(X) has been exhibited by M. Hall [42]. Our approach is somewhat different in that we construct, by means of the free differential calculus, a basis for the dual group Qn* = Hom [QnJ]. The corresponding dual basis of Q. is not the same as the Hall basis, although it bears a superficial resemblance to it. In the course of this construction we re-prove Witt's result [7] that the elements of Xn are just those for which the non-constant terms of the Magnus expansion are all of degree at least n, in short, that the lower central groups coincide with the " dimension groups " of Magnus [2]. Further, we derive a complete set of finite identities for the coefficients in the Magnus expansion of an element of X. The algorithm for Qn(G) is to be found in the last section. The authors wish to thank Julian Brody for his help in simplifying the arguments, and for selection of the example in ?4.

Cohomology Theory of Groups with a Single Defining Relation

Abstract: The Word Problem is that of determining when two elements of F (or two 'words') represent the same element of G, or, equivalently, when a given element W of F lies in R, and so can be written in the form (1). Magnus has solved this problem for the case n = 1, of a single defining relation.3 A complementary problem is that of uniqueness for expressions of the form (1), or more simply that of determining all identities among the relations R,, * **, Rn Xof the form

The Representation of Relational Algebras

Abstract: ?1 Summary A family of binary relations, or sets of ordered pairs, which is a boolean algebra, which is closed under the operations of forming converses and relative products, and which contains an identity relation, constitutes what Tarski has called a proper relational algebra Tarski has given a set of axioms for abstract algebras of this type, and has raised the question whether every abstract algebraic system satisfying these axioms is isomorphic to a proper relational algebra It follows from the results established below that this is not the case In Part One we obtain, after some preliminaries, a set C of conditions which are necessary and sufficient for a finite algebra to be isomorphic to a proper relational algebra A finite model is exhibited which satisfies all of Tarski's axioms, but which does not satisfy all the conditions C, and hence is not representable The conditions C, which are infinite in number and non-finitary in the sense that they contain bound variables, are necessary for any relational algebra, finite or infinite, that is complete as a boolean algebra, to be representable In Part Two it is shown, by means of an infinite model, that no set of finitary axioms suffices to characterize the class of representable relational algebras The relation of these results to some unsolved problems is discussed in the last section (?15)

Phonology and Syntax: The Relation between Sound and Structure

Abstract: A fundamentally new approach to the theory of phonology and its relation to syntax is developed in this book, which is the first to address the question of the relation between syntax and phonology in a systematic way.This general theory differs from its predecessors in the generative tradition in several respects. By arguing that the intonational structure of a sentence determines certain aspects of its stress pattern or rhythmic structure, and not vice versa, it provides a novel view of the intonation-stress relation. It also offers a new theory of the focus-prosody relation that solves a variety of classic puzzles and involves an appeal to the place of a focused constituent in the predicate-argument structure of the sentence. The book also includes other novel features, among them a development of the metrical grid theory of stress (including a complete treatment of English word stress in this framework), the representation of juncture in terms of "silent" positions in the metrical grid (with a treatment of sandhi in terms of this rhythmic juncture), and a "rhythmic" nonsyntactic approach to the basic phonology of function words in EnglishElisabeth 0. Selkirk is Professor of Linguistics at the University of Massachusetts, Amherst. This book is tenth in the series, Current Studies in Linguistics.

An Introduction to Homological Algebra

Abstract: An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author's attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.

On hyperbolic groups

Abstract: We prove that a δ-hyperbolic group for δ < 1 2 is a free product F ∗ G1 ∗ . . . ∗Gn where F is a free group of finite rank and each Gi is a finite group.

C*-Algebras and Finite-Dimensional Approximations

Abstract: Fundamental facts Basic theory: Nuclear and exact $\textrm{C}^*$-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and Kirchberg's factorization property Quasidiagonal C*-algebras AF embeddablity Local reflexivity and other tensor product conditions Summary and open problems Special topics: Simple $\textrm{C}^*$-algebras Approximation properties for groups Weak expectation property and local lifting property Weakly exact von Neumann algebras Applications: Classification of group von Neumann algebras Herrero's approximation problem Counterexamples in $\textrm{K}$-homology and $\textrm{K}$-theory Appendices: Ultrafilters and ultraproducts Operator spaces, completely bounded maps and duality Lifting theorems Positive definite functions, cocycles and Schoenberg's Theorem Groups and graphs Bimodules over von Neumann algebras Bibliography Notation index Subject index.