Algebraic Generalizations of Discrete Grs: A Path to by Benjamin Fine, Gerhard Rosenberger

By Benjamin Fine, Gerhard Rosenberger

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic learn of Fuchsian teams to the extra common context of one-relator items and similar workforce theoretical issues. It offers a self-contained account of yes traditional generalizations of discrete teams.

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5 Bass-Serre Theory The approach we have taken to amalgams in the previous sections has been purely combinatorial. 5 BASS-SERRETHEORY 17 geometric technique for handling group amalgams has been developed by Bass and Serre. It involves the action of a group G on a tree X. By analyzing this action the amalgam structure of G can be deduced. The theory also recovers, in a relatively easy manner, the main theorems of group amalgams- Nielsen-Schreier, Kurosh, Karrass-Solitar. This recovery involves a case by case analysis by amalgamtype.

Let A be an ordered abelian group written additively. 8 GEOMETRICGROUP THEORY: ARBOREALGROUP THEORY 33 archimedean ordered abelian group. In particular all additive subgroups of the reals R are archimedean. If A1 and A2 are ordered abelian groups then the direct sum A1 @A2 can also be made into an ordered abelian group with the lexicographic ordering. However the archimedean property is not necessarily preserved under this construction.. For example under the lexicographic ordering Z ® Z is a non-archimedan ordered abelian group.

Lm+l ! l,,+~. 1,~_~l~ (where l~ ¯ Li if lm¯ Li. Thus without loss of generality we assume that G is countable. Wedefine an ordering on the products of right coset representatives in the L~-1 by taking inverses. Wenow extend this ordering to the set of pairs {g,g-1},g ¯ G, where the notation is chosen so that the leading half of g precedes that of g-~ with respect to ordering _<. Then we set {g,g-1} _< {h,h-~} if either L(g) < L(h) or L(g) = L(h) and the leading half of g strictly precedes that of h, or L(g) = L(h), the leading halves of -~.

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