By A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)
From the reports of the 1st printing of this e-book, released as quantity fifty eight of the Encyclopaedia of Mathematical Sciences:
"... This booklet could be very helpful as a reference and consultant to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The e-book less than evaluate contains monographs on geometric elements of crew idea: Combinatorial crew conception and basic teams" by means of D.J.Collins and H.Zieschang ...: "Some difficulties of workforce conception on the topic of geometry" via R.I.Grigorchuk and P.F.Kurchanov. ... jointly, those articles shape a wide-ranging survey of combinatorial staff conception, with emphasis greatly at the geometric roots of the topic. this may be an invaluable reference paintings for the professional, in addition to offering an outline of the topic for the outsider or amateur. many various themes are defined and explored, with the most effects awarded yet no longer proved. this enables the reader to get the flavor of those themes with no turning into slowed down intimately. either articles provide entire bibliographies, in order that it's attainable to exploit this e-book because the start line for a extra particular learn of a specific subject of curiosity. ... In precis, a really attention-grabbing ebook! Bulletin of the London Mathematical Society, 1996 "... In either essays the authors supply transparent and accomplished definitions, examples and statements (but no longer proofs) of theorems, in order that the publication could be understood by means of a reader with a minimum historical past in workforce idea or geometry. this kind of reader, wanting to determine what's recognized during this sector, will locate this a whole and available shop of information." modern Physics, 1994 "...This survey (Part II) provides for the 1st time that difficulties in monograph shape and incidentally deals a unifying remedy of a number 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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Extra resources for Algebra VII: Combinatorial Group Theory Applications to Geometry
The above is the shortest and most elegant. J. Collins, H. Zieschang The conjugacy problem for F = ( X I - ) is also solvable. 3. Definition and Proposition. A word W == Xfl ... X~n is called cyclically reduced if it is reduced and X~n Xfl is not a peak. Let F = (X I -). (a) Every element of F is conjugate to an element represented by a cyclically reduced word. (b) Two cyclically reduced words represent conjugate elements if and only if one is a cyclic permutation of the other. 13 how to solve the isomorphism problem for free groups.
Furthermore G is unique up to isomorphism. Proof. Existence is demonstrated by choosing, for i E I, disjoint presentations (Xi I Ri ) of the groups G i and systems Yi of words which define generators for the groups K,i(A). Then a presentation ( X I R) for the desired group G is obtained by setting X = UiEI Xi and R = (UiEI Ri)UR' where R' consists of relations which identify corresponding elements of the systems Yi' The uniqueness of G up to isomorphism follows from the uniqueness of the map 'P by the usual category theory argument about the solutions to universal problems.
The method above allows one to determine the subgroups of a given finite index in finitely many steps and can be implemented on computers. It can be modified to a trial-and-error method to determine the index of the subgroup generated by finitely many given elements and to get a presentation - as long as the index is finite. However it is not clear how many steps are needed to determine the index. 7. 13. Definition. Let 6 be a connected 2-complex with Then a covering p : 6 -; C is called universal.
Algebra VII: Combinatorial Group Theory Applications to Geometry by A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)