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 ebook may be very worthwhile as a reference and advisor to researchers and graduate scholars in algebra and and topology." Acta Scientiarum Mathematicarum, Ungarn, 1994 "... The booklet lower than evaluation includes monographs on geometric points of crew conception: Combinatorial crew idea and primary teams" via D.J.Collins and H.Zieschang ...: "Some difficulties of staff thought regarding geometry" by way of 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 can be an invaluable reference paintings for the specialist, in addition to supplying an summary of the topic for the outsider or amateur. many various themes are defined and explored, with the most effects provided yet now not proved. this permits the reader to get the flavor of those issues with out changing into slowed down intimately. either articles supply entire bibliographies, in order that it truly is attainable to take advantage of this e-book because the place to begin for a extra distinctive examine of a selected subject of curiosity. ... In precis, a truly fascinating e-book! Bulletin of the London Mathematical Society, 1996 "... In either essays the authors provide transparent and complete definitions, examples and statements (but no longer proofs) of theorems, in order that the booklet may be understood by means of a reader with a minimum historical past in crew idea or geometry. this type of reader, desiring 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) offers for the 1st time that difficulties in monograph shape and incidentally bargains a unifying remedy of some of the ways 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 info for Algebra VII: Combinatorial Group Theory Applications to Geometry**

**Sample text**

Let 6 be a connected 2-complex with Then a covering p : 6 -; C is called universal. 7r1 (6) 1. A universal covering has the following categorical property: A covering p : 6 -; C is universal if and only if for any other covering p : C -; C with connected total space there is a covering q : 6 -; C such that p = po q. 4 (b), we see that to every 2-complex there exists an universal covering which is uniquely determined up to isomorphism. 14. Examples of Universal Coverings (a) The sphere is the universal cover of the projective plane, see Fig.

Combinatorial Group Theory and Fundamental Groups A product Wi~1 W i:2 ... wtrr , where 10k duced if ik = ik+I implies 10k = Ek+I. = ±1, k = 33 1,2, ... 7. Proposition (a) A reduced product of r elements of a Nielsen reduced system (WI, ... , W n ) of elements has length at least r in the free group. Indeed 1+ IW· I) + r IW"I W"2 ... W"rl > ~(IW 2 IW"IW"2 ... W"rl > IW I 1 <: k <::: r. 1 't2 Zr - 'Z1 22 'lr - 21 'Zk' 'iT' 2 and ~- (b) A Nielsen reduced system (WI' ... ' W n ) of elements of the free group F = ( X I - ) freely generates a free subgroup of F.

Proposition (a) A free group is torsion-free. (b) An abelian subgroup of a free group is cyclic. (c) A free group has trivial centre - except when it is infinite cyclic. 0 To obtain further information on subgroups of a free group, more powerful methods are needed. 5 we have used the method of coverings to deduce that a subgroup of a free group is free and to give a formula for the rank of a subgroup of finite index. Here we give another proof of this result, based on methods of Nielsen [Nielsen 1919].