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

This quantity of the EMS comprises components. the 1st entitled Combinatorial crew conception and primary teams, written by means of Collins and Zieschang, offers a readable and entire description of that a part of crew concept which has its roots in topology within the concept of the basic team and the speculation of discrete teams of modifications. in the course of the emphasis is at the wealthy interaction among the algebra and the topology and geometry. the second one half by way of Grigorchuk and Kurchanov is a survey of modern paintings on teams in terms of topological manifolds, facing equations in teams, really in floor teams and loose teams, a research when it comes to teams of Heegaard decompositions and algorithmic elements of the Poincaré conjecture, in addition to the inspiration of the expansion of teams. The authors have incorporated an inventory of open difficulties, a few of that have no longer been thought of formerly. either elements comprise a variety of examples, outlines of proofs and whole references to the literature. The e-book could be very worthwhile as a reference and consultant to researchers and graduate scholars in algebra and topology.

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

**Example text**

8 and let H < G, [G : H] < co. 8. Moreover, p(H) = (Riemann-Hurwitz formula). [G : HI . 2 (c) since in this case p(G) = 4g - 4 = -2 . x(C), p(H) = -2 . x(C’). This argument can be generalized to the other groups G using branched coverings. 221. 0 76 I. J. Collins, H. Zieschang Consider a closed surface C and a group r of symmetries of the complex C, that is automorphisms of C. Then we obtain a mapping p’: C + C/r. The mappings of r can be lifted to the universal covering p: E --+ C and we obtain a discontinuous group G acting on IE.

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.

If G and G’ are geometrically isomorphic, then the surfaces ItX’/G’, IE/G have the same genus (g’ = g) and the “branching properties” are the “same”. 10. Theorem. If two planar discontinuous groups are isomorphic then they are also geometrically isomorphic. Proof. A nice geometric argument shows that the elements of finite order are conjugate to the powers of the s,, that the subgroups (si) are maximal I. J. Collins, H. Zieschang 70 finite subgroups and no two are conjugate. So the numbers hl, .