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

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

"... This publication can be very helpful as a reference and consultant to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The ebook less than evaluation involves monographs on geometric features of crew conception: Combinatorial crew conception and basic teams" by means of D.J.Collins and H.Zieschang ...: "Some difficulties of workforce conception concerning geometry" by means of R.I.Grigorchuk and P.F.Kurchanov. ... jointly, those articles shape a wide-ranging survey of combinatorial staff idea, with emphasis greatly at the geometric roots of the topic. it will be an invaluable reference paintings for the specialist, in addition to delivering an outline of the topic for the outsider or beginner. many alternative subject matters are defined and explored, with the most effects awarded yet now not proved. this permits the reader to get the flavor of those themes with out changing into slowed down intimately. either articles supply finished bibliographies, in order that it truly is attainable to take advantage of this booklet because the start line for a extra unique examine of a selected 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 no longer proofs) of theorems, in order that the ebook might be understood by way of a reader with a minimum heritage in workforce thought or geometry. any such reader, wanting to determine what's recognized during this quarter, will locate this a whole 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 remedy of a few of the ways to their ideas, 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**

**Example text**

Theorem. Let the group G act without inversion on the tree X and let Y = XjG be the quotient graph. For any vertex v and edge a of Y let fJ and jj denote a vertex and edge of X lying over v and a respectively. Then G is (isomorphic to) the fundamental group of a graph (g, Y) of groups whose vertex groups and edge groups, respectively, are stabilisers of the form Gv = Stabc(fJ) and Ga = Stabc(jj). 22. Theorem. Let G be the fundamental group of a graph (9, Y) of groups and E+ (Y) an orientation of Y.

PI(U)! Pi",PI(U)! Pj-I" ,PI(U)! Pj .. ,PI(U)! , (c) IPk-l .. ·PI(U)! Pk .. ·PI(U)! , l~i~p, p+1~j~q, q+1~k~n. 0 The inequalities (a), (b) and (c) are illustrated by the following diagram in which length is plotted upwards. 6 is derived as follows. J. Collins, H. Zieschang 50 u \ ~~ l . -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.

We will explain it with an example. 12. Example. 6 (d). Then 7r1(N2) = (a, b I a2b2). The symmetric group Sym(3) consists of the six elements 1, r = (12), s = (13), t = (23), u = (123), v = (132), where r, s, t have order 2 and u, v have order 3. Elements f(a), f(b) with f(a)2 f(b)2 = 1 determine a homomorphism f : 7r1(N2) -; Sym(3). If f(a) = 1 or f(b) = 1 then f is not transitive. If f(a) has order 2 then, up to equivalence, f(a) = r, f(b) = s. If the order of f(a) is 3 then f(b) = f(a)-I and, up to equivalence, f(a) = u, f(b) = v.