By Benjamin Fine, Gerhard Rosenberger

ISBN-10: 0585418365

ISBN-13: 9780585418360

ISBN-10: 0824703197

ISBN-13: 9780824703196

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic examine of Fuchsian teams to the extra basic context of one-relator items and comparable staff theoretical issues. It presents a self-contained account of definite normal generalizations of discrete teams.

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**Additional info for Algebraic Generalizations of Discrete Grs: A Path to Combinatorial Gr Theory Through One-Relator Products**

**Example text**

In Serre’s original work on groups acting on trees [Se], certain trees were constructed from discrete valuations and Morgan and Shalen’s construction can be considered a generalization of this. A complete discussion of A-trees can be found in the survey articles by Morgan[Mo],Shalen [Sh 1],[Sh 2], the work of Bass [B]and Alperin and Bass [A-B] or the papers of Chiswell [Ch 3],[Ch 4],[Ch 5]. Let A be an ordered abelian group written additively. 8 GEOMETRICGROUP THEORY: ARBOREALGROUP THEORY 33 archimedean ordered abelian group.

Zieschang in [C-Z 1] which we do not consider here in detail. Werestrict ourselves in these notes to the free product of two groups with an amalgamated subgroup, although the method works more generally. Let G = H1 *A H~, H1 ¢ A ¢ H:, denote the non-trivial free product of the groups H~ and H~ with the amalgamated subgroup A -- H~ ~ Hu. If A = {1) then G is just the free product G = H1 * Hu of H~ and H:. Wechoose in each H~, i = 1, 2, a system Li of left coset representatives of A in Hi normalized by taking 1 to represent A.

Wewill discuss these classes of groups in the next chapter. From an algebraic point of view, one-relator groups are of interest because they provide a natural generalization Of free groups, with which they exhibit some similarities and further because they lend themselves to a general treatment. The article by G. Baumslag[GB-1] and the sections in the book of Lyndon and Schupp [L-S] provide more background in the general interest in one-relator group theory. Dehn and then proved by W. Magnusin 1929 [M 1].

### Algebraic Generalizations of Discrete Grs: A Path to Combinatorial Gr Theory Through One-Relator Products by Benjamin Fine, Gerhard Rosenberger

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