# 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 elements. the 1st entitled Combinatorial crew thought and primary teams, written by means of Collins and Zieschang, offers a readable and entire description of that a part of crew conception which has its roots in topology within the concept of the basic crew and the idea of discrete teams of alterations. through 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 contemporary paintings on teams in terms of topological manifolds, facing equations in teams, rather in floor teams and unfastened teams, a examine when it comes to teams of Heegaard decompositions and algorithmic elements 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 no longer been thought of formerly. either elements comprise various examples, outlines of proofs and entire references to the literature. The booklet might be very valuable as a reference and consultant to researchers and graduate scholars in algebra and topology.

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

**Example text**

Moreover, the relation between algebraic und geometric properties has to be considered. 9. (a) There is an (algebraic) isomorphism between the groups. (b) There is a geometric isomorphism between the groups, that is there are realizations (E, G) and (IE’, G’) and an isomorphism h: II? + IE such that x H hh’xh defines an isomorphism from G to G’. Clearly, a geometric isomorphism is algebraic. Classification with respect to geometric isomorphism is easily carried out. 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”.

Hence there is an epimorphism from the group G = (a, b 1 [a, b13 = 1) to H given by a H ~1, b H x2. However one can lift back the decomposition of H to give a non-trivial decomposition of G as an amalgamated free product. Similar arguments apply for G = (a, b 1 [a, b12’“+’ = 1) whence it follows that if m is not a power of two, then G = (a, b 1 [a, blm = 1) also has a non-trivial decomposition. It is an open question whether or not G = (a, b I [a, b12” = 1) has such a decomposition - although since the relator has exponent sum 0 on both a and b, G is an HNN-extension in various ways.

21. 16. 5. 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.