Title | Algebra VII [electronic resource] : Combinatorial Group Theory Applications to Geometry / edited by A. N. Parshin, I. R. Shafarevich |
---|---|

Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1993 |

Connect to | http://dx.doi.org/10.1007/978-3-642-58013-0 |

Descript | VII, 242 p. online resource |

SUMMARY

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 fรผr Mathematik, 1995

CONTENT

I. Combinatorial Group Theory and Fundamental Groups -- 1. Group Presentations and 2-Complexes -- 2. Free Groups and Free Products -- 3. Surfaces and Planar Discontinuous Groups -- 4. Cancellation Diagrams and Equations Over Groups -- 5. 3-Manifolds and Knots -- 6. Cohomological Methods and Ends -- 7. Decision Problems -- II. Some Questions of Group Theory Related to Geometry -- 1. Equations in Groups and Some Related Questions -- 2. Splitting Homomorphisms and Some Problems in Topology -- 3. On the Rate of Growth of Groups and Amenable Groups -- Author Index

Mathematics
Group theory
Algebraic topology
Mathematics
Group Theory and Generalizations
Algebraic Topology