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

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

"... This e-book might be very important as a reference and consultant to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The ebook lower than evaluate includes monographs on geometric features of workforce concept: Combinatorial crew concept and basic teams" by way of D.J.Collins and H.Zieschang ...: "Some difficulties of staff thought concerning geometry" by way of R.I.Grigorchuk and P.F.Kurchanov. ... jointly, those articles shape a wide-ranging survey of combinatorial staff idea, with emphasis a great deal at the geometric roots of the topic. this can be an invaluable reference paintings for the specialist, in addition to delivering an outline of the topic for the outsider or beginner. many various subject matters are defined and explored, with the most effects offered yet no longer proved. this enables the reader to get the flavor of those subject matters with out changing into slowed down intimately. either articles provide accomplished bibliographies, in order that it's attainable to exploit this e-book because the place to begin for a extra specified research of a selected subject of curiosity. ... In precis, a truly attention-grabbing publication! Bulletin of the London Mathematical Society, 1996 "... In either essays the authors provide transparent and accomplished definitions, examples and statements (but now not proofs) of theorems, in order that the publication should be understood through a reader with a minimum heritage in team idea or geometry. this kind of reader, desiring to determine what's recognized during this quarter, 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 deals a unifying therapy of some of the techniques 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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**Additional info for Algebra VII: Combinatorial Group Theory Applications to Geometry**

**Sample text**

18-19 we have seen ways to construct the fundamental group of a complex from the fundamental groups of subcomplexes. The group-theoretic analogue of this procedure consists of certain ways of building up groups from subgroups. This ultimately leads to the notion of the fundamental group of a graph of groups which we shall describe below. We begin, however, with a discussion of some important special cases. I. 1. Definition. Let (Gi)iEI be a family of groups and (K,i : A -> Gi)iEI a family of embeddings of the fixed group A in the various groups Gi .

Every countable group H can be embedded in a group G generated by two elements. Proof. Let H be a countable group and let the elements of H be enumerated as 1 = h o, hI, h 2 , .... Let Go be the free product H * F where F is the free group with basis {a, b}. Let Al = (b-nab n : n = 0,1,2, ... ) and A2 = (hna-nba n : n = 0,1,2, ... ). An easy cancellation argument shows that Al is free on the given generators (the generators actually form a Nielsen 1. Combinatorial Group Theory and Fundamental Groups 39 reduced system) and projecting from Go to F shows that the same holds for A 2.

Orphism f : 7r1 (C, v2-; Sym( n) there is an n-sheeted connected covering p : C -; C such that S (c, p) = (I). These results permit the effective determination of the n-sheeted coverings over a given complex C. 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).