By A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

ISBN-10: 3540637044

ISBN-13: 9783540637042

ISBN-10: 3642580130

ISBN-13: 9783642580130

From the studies of the 1st printing of this publication, released as quantity fifty eight of the Encyclopaedia of Mathematical Sciences:

"... This ebook should be very precious as a reference and advisor to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The e-book below overview comprises monographs on geometric features of workforce concept: Combinatorial team conception and basic teams" by means of D.J.Collins and H.Zieschang ...: "Some difficulties of team concept concerning geometry" through R.I.Grigorchuk and P.F.Kurchanov. ... jointly, those articles shape a wide-ranging survey of combinatorial workforce concept, with emphasis a great deal at the geometric roots of the topic. it will be an invaluable reference paintings for the specialist, in addition to supplying an outline of the topic for the outsider or amateur. many alternative subject matters are defined and explored, with the most effects awarded yet no longer proved. this enables the reader to get the flavor of those issues with no turning into slowed down intimately. either articles provide entire bibliographies, in order that it's attainable to exploit this publication because the place to begin for a extra distinctive learn of a specific subject of curiosity. ... In precis, a really fascinating e-book! Bulletin of the London Mathematical Society, 1996 "... In either essays the authors provide transparent and entire definitions, examples and statements (but now not proofs) of theorems, in order that the e-book may be understood by means of a reader with a minimum historical past in team conception or geometry. this type of reader, desiring to determine what's recognized during this region, will locate this a whole and available shop of information." modern Physics, 1994 "...This survey (Part II) offers for the 1st time that difficulties in monograph shape and incidentally deals a unifying remedy of a few of the techniques to their strategies, so far as they're identified, including tricks to open difficulties. A titbit for each reader!" Monatshefte für Mathematik, 1995

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**Additional resources for Algebra VII: Combinatorial Group Theory Applications to Geometry**

**Sample text**

J. Fig. 1 many Whitehead automorphisms will replace u and v by minimal words Uo and Vo. 9, Uo can be transformed into Vo by a sequence of applications of permutation and Whitehead automorphisms in such a way that the lengths of the intermediate words obtained are the same as the length of Uo and Vo. This means that the number of automorphisms in the sequence can be bounded in terms of the length of Uo and the result follows. We illustrate the theorem with examples. 10. Example. Let X = {s, t}.

19 for definitions). 28. Definition and Examples (a) Let G be a group and X, Y two systems of elements of G. Then X and Yare Nielsen equivalent if one can be transformed into the other by a finite sequence of applications of the operations (cf. 8) (TO) permute the elements of the system; (T1) replace an element by its inverse; (T2) replace an element x E X by the product xx', where x' E X, x' -I- x. (b) Let G = (s, t I [s, t] = 1). Then any system {x, y} of generators of Gis Nielsen equivalent to {s, t}.

Theorem. Let F = ( X sented by a unique reduced word. I - ). Then each element of F is repre- Proof. Let Wo be the set of all reduced words over X. 3 it suffices to define an action for each X E X. Given W == X~l ... W XX~l .. l = U and hence exactly one reduced word represents g. 1 justifies defining the length Igl relative to X of an element of F as the length of the unique reduced word representing g. 2. Remark. 1 are available - see [Cohen 1989]. The above is the shortest and most elegant. J.

### Algebra VII: Combinatorial Group Theory Applications to Geometry by A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

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