# 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 includes elements. the 1st entitled Combinatorial workforce conception and primary teams, written through Collins and Zieschang, offers a readable and finished description of that a part of staff idea which has its roots in topology within the idea of the elemental crew and the idea of discrete teams of variations. 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 contemporary paintings on teams on the subject of topological manifolds, facing equations in teams, really in floor teams and loose teams, a examine by way of teams of Heegaard decompositions and algorithmic points of the Poincaré conjecture, in addition to the concept 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 include quite a few examples, outlines of proofs and whole references to the literature. The e-book may be very invaluable as a reference and consultant to researchers and graduate scholars in algebra and topology.

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

Example text

For the full determination of the possible symmetry groups on a torus see [Zieschang 1981, Chap. 21, [ZVC 1988, Chap. 81. H. 4. Theorem. Every discontinuous group of the plane mental group of a surface as a subgroup offinite index. e. of the form G = (sl,. . u,]). Tl le e1ements of finite order are conjugate to the powers of the si. So it suffices to find a homomorphism ‘p of G to a finite group E such that I has order hi, 1 < i 5 m. 101). 5. Selberg Lemma. Any finitely generated subgroup of GL(m,C) contains a torsion free subgroup of finite index.

Using Tietze transformations, one can introduce new generators bb = b , bl = abaa’, b2 = a2baa2, cj = ajca-j (j E Z). ‘1c~ = 1, ab,a-’ = b,+l, acja-l = cj+1 ) where i = 0,l and j E Z. J. Collins, H. ,c~) are free on the displayed generators and so G is given as an HNN-extension with Go as base group and a as stable letter. It follows immediately that {b, c} (= {bo, co}) is a basis for a free subgroup of G. To seethat {a, c} is also a basis for a free subgroup, note firstly that any reduced word W giving a relation over {a, c} must have zero exponent sum in a since it is a consequence of the original relator R.

In these cases the groups obtained are either trivial or can be realized by groups of motions of the sphere S’. The groups of the last form with rotation orders (hl, h2. h3) are the dihedral groups D, of order 2n (case (2,2, n)) and the platonic groups: tetrahedral group (2,3,3) of order 12, octahedral group (2,3,4) of order 24, dodecahedral group (2,3,5) of order 60. 12. Geometric Approach. We will illustrate the geometric proof of existence with the simplest examples, namely the triangle groups where g = 0, m = 3: Construct a triangle ABC with sides a, b, c and angles E, %, F; this can be done in the euclidean plane if 5 + % + T = 7rIT, on the sphere if the sum is bigger than 7r and in the Bolyai-Lobachevskij (hyperbolic) plane if the sum is smaller than 7r.