# Get Algebraic Generalizations of Discrete Grs: A Path to 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 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 yes common generalizations of discrete teams.

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**Example text**

From this it is straightforward that if X is Nielsen reduced then < X > is free with X as a basis. Hence every finitely generated subgroup of a free group is free. This was Nielsenâ€™s original proof. It can be extended to remove the finitely generated restriction {see [L-S]}. ,x~}, 1 < n < m, is Nielsen reduced and generates F then ra = n and X is a basis for F. ,y~ are other free generators of < X > and both sets are ordered according to the length L then L(y~) >_ L(x~), 1,.. , n. It is clear that a Nielsen transformation applied to a basis of a finitely generated free group defines an automorphism.

From the the existence of the Dehn algorithm we get the following corollaries. 1. A hyperbolic group G is t~ni~ely presented. 2. A hyperbolic 29 group G has a solvable word problem. In fact if G is 5-hyperbolic with generating set X and if we define R = {W E F(X);L(W) <_ 85 and W-- 1 in then < XIR > is a Dehn presentation for G, that is has a Dehn algorithm. Gromov [Gr] further has proved that hyperbolic groups have solvable conjugacy problem while Sela [Sel 1] has shown that the isomorphism problem is solvable for the class of torsion-free hyperbolic groups.

Karrass and Solitar have also proven a corresponding subgroup theorem for HNNgroups. 2. , K; rels K,t~lL~t~ = f~(L~),i 1,... > . Then any subgroup H of G is a treed HNNgroup. Further the vertex groups of the base of H are conjugates of the base K intersected with H while the amalgamated subgroups in the base of H are conjugates of the Li intersected with H and the non-trivial associated subgroups in H are conjugates of K or of the Li intersected with H. 5 Bass-Serre Theory The approach we have taken to amalgams in the previous sections has been purely combinatorial.

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

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