Algebra Seven: Combinatorial Group Theory. Applications to by Parshin A. N. (Ed), Shafarevich I. R. (Ed)

By Parshin A. N. (Ed), Shafarevich I. R. (Ed)

This quantity of the EMS comprises components. the 1st entitled Combinatorial staff idea and primary teams, written by way of Collins and Zieschang, offers a readable and entire description of that a part of team concept which has its roots in topology within the concept of the elemental team and the speculation of discrete teams of variations. in the course of the emphasis is at the wealthy interaction among the algebra and the topology and geometry. the second one half via Grigorchuk and Kurchanov is a survey of contemporary paintings on teams in terms of topological manifolds, facing equations in teams, really in floor teams and unfastened teams, a learn by way of teams of Heegaard decompositions and algorithmic features of the Poincaré conjecture, in addition to the thought of the expansion of teams. The authors have incorporated a listing of open difficulties, a few of that have now not been thought of formerly. either elements include a variety of examples, outlines of proofs and whole references to the literature. The e-book can be very helpful as a reference and advisor to researchers and graduate scholars in algebra and topology.

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Extra resources for Algebra Seven: Combinatorial Group Theory. Applications to Geometry

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Corollary. 1. 6. Theorem. The conjugacy problem for sion is solvable. •i a one-relator group with tor- The same general method and the torsion theorem for HNN-extensions provide a satisfactory account of torsion in one-relator groups. Until recently little was known in general for the conjugacy problem for one-relator groups without torsion. 11) may lead to a solution. The idea of applying geometric methods to one-relator groups was introduced by Lyndon who used cancellation diagrams to give an alternative proof of the Freiheitssatz.

Proposition. Let Go = (al, bl,. ,ag, b, ] [al, bl] . . [a,, bg]). Then G = (X I R) is isomorphic to Go if and only if IX] = 29 and there is an isomorphism of the two free groups involved carrying [al, bl] . . [ag, b,] to R*l. ai ). 0 One further positive result is due to Pride [Pride 19771. 15. Proposition. Let G = ( a, b I R ) where R E S”, with m > 2, and 5’ is not a primitive. Then for any pair (x, y) of generators of G there exist words U and V of F(a, b) representing x and y and an automorphism of F(a, b) carrying (U, V) to (a, b).

Classification (a) Any finite surface plexes S,>,, Ng,r. J. Collins, H. Zieschang I. Combinatorial (b) If an orientable or non-orientable compact surface has genus g and r boundary components then its fundamental group is isomorphic to T dSg,r) = (a,. . ,ST,tl,Ul,. . ,tg,ug 9 I nsi i=l T rrl(N,,,) j=l 9 = (sl, . . , s,, vl, . . , vg 1n si n vj), i=l (cl ffl (Sg,,) = ;;;+F-l { ; ; ;> ; z2 = respectively. j=l ;; { Hl(Ng,,) @r n[tj,rrjl) $ ;zg+r-1 E-l ifr = 0, ifr>O. 0 If r > 0 then by Tietze transformations one of the generators and the single defining relation can be omitted and hence the fundamental group is free of rank 2g + r - 1 or g + r - 1, respectively.

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