By A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)
From the experiences of the 1st printing of this ebook, released as quantity fifty eight of the Encyclopaedia of Mathematical Sciences:
"... This publication should be very worthwhile as a reference and advisor to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The publication lower than evaluate includes monographs on geometric elements of crew idea: Combinatorial team conception and primary teams" via D.J.Collins and H.Zieschang ...: "Some difficulties of crew conception with regards to geometry" through R.I.Grigorchuk and P.F.Kurchanov. ... jointly, those articles shape a wide-ranging survey of combinatorial crew idea, 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 beginner. many various themes are defined and explored, with the most effects awarded yet now not proved. this enables the reader to get the flavor of those themes with out changing into slowed down intimately. either articles provide complete bibliographies, in order that it's attainable to take advantage of this e-book because the start line for a extra particular examine of a specific subject of curiosity. ... In precis, a really fascinating publication! 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 via a reader with a minimum historical past in workforce conception or geometry. the sort of reader, wanting to determine what's recognized during this sector, will locate this an entire and obtainable shop of information." modern Physics, 1994 "...This survey (Part II) provides for the 1st time that difficulties in monograph shape and incidentally bargains a unifying therapy of many of the methods to their suggestions, so far as they're recognized, including tricks to open difficulties. A titbit for each reader!" Monatshefte für Mathematik, 1995
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From the stories of the 1st printing of this e-book, released as quantity fifty eight of the Encyclopaedia of Mathematical Sciences:". .. This ebook could be very worthwhile as a reference and advisor to researchers and graduate scholars in algebra and and topology. " Acta Scientiarum Mathematicarum, Ungarn, 1994 ". .
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Extra resources for Algebra VII: Combinatorial Group Theory Applications to Geometry
C) Let X be any finite tree. Then for any graph (Q, X) of groups, 7r1 (Q, X, v) can be constructed by repeated formation of amalgamated free products. If now we assume X is infinite (but every vertex has finite degree), then it is an ascending union of finite subtrees Xn and 7r1(Q,X,V) is the ascending union of the corresponding fundamental groups 7r1 (Qn, Xn). Sometimes 7r1 (Q, X, v) is called a tree product in these circumstances. If (Q, X) is an arbitrary graph of groups, then 7r1(Q,X,V) can be regarded as an HNNextension by a set of stable letters corresponding to edges not in a maximal tree T over the base group which is the tree product defined by the restriction of (9,X) to the tree T.
Cn) and the result follows. 5 can be derived by a similar argument. In both cases the integer n is referred to as the length (relative to the given decomposition of G) of the element g. 6. Corollary. The natural maps embed the factors of a free product with amalgamation and the base group of an HNN-extension. 7. Examples. There are numerous examples of amalgamated free products and HNN-extensions some of which we have already encountered. (a) Let G I = (al,b l I -) and G 2 = (a2,b 2 I -). If Al = (aIbIallbll) and A2 = (b 2a 2b2' l a2'l) then Al and A2 are infinite cyclic subgroups of free I.
The natural maps embed the factors of a free product with amalgamation and the base group of an HNN-extension. 7. Examples. There are numerous examples of amalgamated free products and HNN-extensions some of which we have already encountered. (a) Let G I = (al,b l I -) and G 2 = (a2,b 2 I -). If Al = (aIbIallbll) and A2 = (b 2a 2b2' l a2'l) then Al and A2 are infinite cyclic subgroups of free I. Combinatorial Group Theory and Fundamental Groups 37 Fig. 14 (c)). If, on the other hand we put Go = ( b1, a2, b2 I - ), Al = (b2a252'la2'lb1) and A2 = (b 1) then Al and A2 are infinite cyclic subgroups of the same group Go, and, if we write t for aI, then G = ( Go, t I r1 A1t = A2 ) is also this fundamental group.
Algebra VII: Combinatorial Group Theory Applications to Geometry by A. N. Parshin (auth.), A. N. Parshin, I. R. Shafarevich (eds.)