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

This quantity of the EMS includes components. the 1st entitled Combinatorial staff thought and primary teams, written through Collins and Zieschang, offers a readable and entire description of that a part of team idea which has its roots in topology within the conception of the basic staff and the speculation 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 through Grigorchuk and Kurchanov is a survey of contemporary paintings on teams with regards to topological manifolds, facing equations in teams, quite in floor teams and unfastened teams, a research by way of teams of Heegaard decompositions and algorithmic facets 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 a variety of examples, outlines of proofs and entire references to the literature. The publication should be very beneficial 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**

J~~}U{~~~,~f~:l~j~g} (or {p,“, uF1 : 1 < j < r} U {u,“’ : 1 5 j < g}, respectively,) and one pair of faces pkl with the following boundary conditions: the rj, p~j,vj start and end at v, uj runs from v to vj, ,oj from vj to vj, and dp = fiUjpjU;l . 2. 3. Examples: Closed Surfaces. 5. , rs, ~L~‘,T;‘,P~ and is closed. The Euler characteristic is x(Sg) = 2 - 29. Hence, S, and Sh with g # 1~are not homeomorphic. The number g is called the genus of S,. The surface S, is orientable. In Ng the star around v is v[‘, ~1, Z&Y’,24, .

6. Theorem. g. ai and ai) are G-equivalent. 0 Suppose that g(D) n D is l-dimensional. Then there are two possibilities: (a) g(D) nD consists of one edge cr E lE. Then g-l(D) n D = g-l(o) # of1 and gP1( o ) is t h e only other edge in D which is G-equivalent to O. (b) g(D) n D consists of two edges cr, r E IE and the two edges have a common vertex v. Then gE(a) = r, with E = *l, and g is a rotation with rotation centre V. In particular, g has finite order. In both cases we obtain a pair of G-equivalent edges in the boundary of D.

FiSi'fi[tj>Uj]). i=l j=l 0 The assertion about the subwords of relators is claimed only for planar discontinuous groups, not for all groups given by a presentation of the above form. The generators denoted by Si represent mappings of finite order which fix some vertex, thus, behave like a rotation. For a planar discontinuous group G neither the form of its fundamental domain D nor its presentation are invariants and there arises the question of classifying planar groups. Another problem is whether a given presentation (or formal form of a fundamental domain) can be realized by a planar discontinuous group.

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

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