By Benjamin Fine, Gerhard Rosenberger
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 common context of one-relator items and comparable workforce theoretical issues. It offers a self-contained account of convinced common 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
A subgroup of a finitely generated group is quasiconvex if the vertices in the subgroup form a quasiconvex set in the Cayley graph. 5. perbolic. A quasiconvex subgroup of a hyperbolic group is hy- Thinking of the Cayley graph of a free group as a tree it is clear that a finitely generated subgroup of a finitely generated free group is quasiconvex. 30 ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS Further an infinitely generated subgroup is not hyperbolic and hence not quasiconvex. Therefore we have.
It has also been shownthat all polycyclic-by-finite groups as well as finitely generated free extensions of finitely generated free groups are in this class. 1. Let G be a tinitely generated group with a complete rewriting system A. Then G is finitely presented and the set of A-reduced words forms a regular language. Further the word problem is solvable. Rewrite rules need not shorten the length of a word. However we do get the following interesting tie with both free groups and hyperbohc groups.
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.
Algebraic Generalizations of Discrete Grs: A Path to Combinatorial Gr Theory Through One-Relator Products by Benjamin Fine, Gerhard Rosenberger