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

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 learn of Fuchsian teams to the extra common context of one-relator items and comparable team theoretical issues. It presents a self-contained account of yes traditional generalizations of discrete teams.

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

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A geodesic in the Cayley graph is a path between two points with minimal length relative to the word metric. A geodesic triangle is a triangle with geodesic sides. A geodesic triangle in F is 5-thin if any point on one side is at a distance less than 5 from some point on one of the other two sides. F is 5-hyperbolic if every geodesic triangle is 5-thin. Finally G is word-hyperbolic or just hyperbolic if G is 5-hyperbolic with respect to some generating set X and some fixed 5 _> 0. Gromovfurther showed that being hyperbolic is independent of the generating set although the 5 may change, that is if G is hyperbolic with respect to one finite generating set it is hyperbolic with respect to all finite generating sets.

Wedo an induction on the length of the relator R. If R involves only one generator, the theorem holds. To avoid using multiple indices, suppose G =< a,... , c, t; R = 1 > where t = x,~ renaming the generators without indices. Suppose first that t has exponent sum zero in the relator R. For any generator g, let g~ = t~gt -~, i E Z.

1. (Rips Theorem) A finitely generated ~-~ee group a free produc~ of finitely many finitely generated free abelian groups and surface groups. Morganand Shalen [Mo-S1] further extended the concept of an ll~-tree to a A-tree where A is an arbitrary ordered abelian group. This concept arose in their paper from an example of valuation on a field where the ordered abelian group is the valuation group. 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.

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

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