Collins, D.J., Grigorchuk, R.I., Kurchanov, P.F., Zieschang, H.
Kostrikin, A.I., Shafarevich, I.R. (Eds.)
Translated by Cohn, P.M.
Originally published as volume 58 in the series: Encyclopaedia of Mathematical Sciences
1993, VII, 242 p.
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From the reviews of the first printing of this book, published as volume 58 of the Encyclopaedia of Mathematical Sciences: "... This book will be very useful as a reference and guide to researchers and graduate students in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The book under review consists of two monographs on geometric aspects of group theory: Combinatorial group theory and fundamental groups" by D.J.Collins and H.Zieschang ...: "Some problems of group theory related to geometry" by R.I.Grigorchuk and P.F.Kurchanov. ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest. ... In summary, a very interesting book! Bulletin of the London Mathematical Society, 1996 "... In both essays the authors give clear and comprehensive definitions, examples and statements (but not proofs) of theorems, so that the book can be understood by a reader with a minimal background in group theory or geometry. Such a reader, needing to find out what is known in this area, will find this a full and accessible store of information." Contemporary Physics, 1994 "...This survey (Part II) presents for the first time that problems in monograph form and by the way offers a unifying treatment of the various approaches to their solutions, as far as they are known, together with hints to open problems. A titbit for every interested reader!" Monatshefte für Mathematik, 1995
I. Combinatorial Group Theory and Fundamental Groups.- 1. Group Presentations and 2-Complexes.- § 1.1. Presentations of Groups.- § 1.2. Complexes and Fundamental Groups.- § 1.3. Subgroups and Coverings.- 2. Free Groups and Free Products.- § 2.1. Free Groups.- § 2.2. Amalgamated Free Products and Graphs of Groups.- § 2.3. Automorphisms of Free Groups.- § 2.4. One-Relator Groups.- 3. Surfaces and Planar Discontinuous Groups.- § 3.1. Surfaces.- § 3.2. Planar Discontinuous Groups.- § 3.3. Subgroups of Planar Groups.- § 3.4. Automorphisms of Fuchsian Groups.- § 3.5. Relations to Other Theories of Surfaces.- 4. Cancellation Diagrams and Equations Over Groups.- § 4.1. Cancellation Diagrams.- § 4.2. Locally Indicable Groups and Equations Over Groups.- 5. 3-Manifolds and Knots.- § 5.1. Fundamental Groups of 3-Manifolds.- § 5.2. Haken Manifolds.- § 5.3. On Knots and Their Groups.- 6. Cohomological Methods and Ends.- § 6.1. Group Extensions and Cohomology.- § 6.2. Ends of Groups.- 7. Decision Problems.- § 7.1. Decision Problems and Algorithms.- § 7.2. Unsolvable Decision Problems.- § 7.3. Automata and Groups.- Index of Notation.- II. Some Questions of Group Theory Related to Geometry.- 1. Equations in Groups and Some Related Questions.- § 1. Basic Concepts and the Theorem of Makanin.- § 2. Solutions of Systems and Homomorphisms.- § 3. Fundamental Sequences and Razborov’s Theorem.- § 4. On the Structure of the Set of Solutions of Quadratic Equations in Free Groups.- § 5. Coefficient-Free Quadratic Equations.- § 6. The Classification of Epimorphisms from Surface Groups to Free Groups.- § 7. On the Minimal Number of Fixed Points in the Homotopy Class of Mappings and the Width of Elements in Free Groups.- § 8. On Quadratic Equations in Hyperbolic Groups.- 2. Splitting Homomorphisms and Some Problems in Topology.- § 1. Heegaard Decompositions of 3-Manifolds and their Equivalence.- § 2. The Poincaré Conjecture and Three Algorithmic Problems Connected with 3-Manifolds.- § 3. Information on Aut ?1(T) and Some of its Subgroups and Factor Groups.- § 4. On the Problem of the Equivalence of Splitting Homomorphisms.- § 5. On an Algebraic Reduction of the Poincaré Conjecture and the Algorithmic Poincaré Problem.- § 6. Some Analogues with the Group of Symplectic Matrices and the Torelli Group.- § 7. Algebraic Reduction of the Problem of the Equivalence of Links.- § 8. On the Andrews-Curtis Conjecture.- 3. On the Rate of Growth of Groups and Amenable Groups.- § 1. On the Growth of Graphs and of Riemannian Manifolds.- § 2. On the Notion of Growth of a Finitely Generated Group.- § 3. On the Proof of Gromov’s Theorem and Some Related Results.- § 4. Example of a Group of Intermediate Growth and the Construction Scheme of such a Group.- § 5. On the Structure of the Set of Growth Degrees of Groups that are Residually-p Groups.- § 6. On an Application of the Theory of Groups of Polynomial Growth to Geometry.- § 7. Regularly Filtered Surfaces and Amenable Groups.- Index of Notation.- Author Index.