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

This quantity of the EMS comprises elements. the 1st entitled Combinatorial staff idea and basic teams, written by means of Collins and Zieschang, presents a readable and accomplished description of that a part of team concept which has its roots in topology within the concept of the elemental staff and the idea of discrete teams of differences. during the emphasis is at the wealthy interaction among the algebra and the topology and geometry. the second one half by means of Grigorchuk and Kurchanov is a survey of modern paintings on teams in terms of topological manifolds, facing equations in teams, quite in floor teams and unfastened teams, a examine by way of 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 now not been thought of formerly. either elements include a variety of examples, outlines of proofs and entire references to the literature. The ebook might be very helpful 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**

**Sample text**

2. Planar Discontinuous Groups The universal covering of a surface is a simply connected surface. If we consider only surfaces without boundary it turns out that there are only two types of simply connected surfaces: the orientable closed surface of genus 0 and the plane ~ as long as we are only concerned with topological, (real) differentiable or combinatorial properties of the surface. 11. 1. SchSnflies Theorem. Let S be a simply connected surface without boundary and y c S a simple closed path.

10 were given in [Zieschang 19701,[McCool-Pietrowski 19711. 8. Theorem. Let G = (X / R). (a) If 1x1 2 3 , th en G has trivial centre. (b) If IX = 2 , and G is not abelian then the centre of G is trivial or infinite cyclic. 12. Example. Let F = F(s, t) be free with basis (s, t) and let u = s3t5 and u = s3t3s3t2. 10, u and vi1 are not equivalent under Aut F. Now consider the one-relator group G = (s, t I s3t5). We introduce a new generator z = t2 and perform the following Tietze transformations which give presentations of the same group G: G = (s, t,z 1zt-2, We indicate below an elementary argument that specifies in terms of the relator when G = (X 1 R) is abelian.

J. Collins, H. Zieschang I. e. is induced by a homeomorphism of a surface with boundary, then rank(Fix(a)) 5 rank(F). e. 12 was exploited to prove that if o is a periodic automorphism of F, then Fix(o) is a free factor of F. The full result was first established in [Gersten 1983,1987]. The Whitehead method can be generalized to free products of groups to obtain a peak-reduction lemma as well as a result on the stabilizer of an element; see [Collins-Zieschang 1984,1987]. 18. Theorem. Let F be a free group of finite rank.

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

by Paul

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