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 normal context of one-relator items and similar staff theoretical issues. It offers a self-contained account of sure typical generalizations of discrete teams.

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Let G = G1 *A G2 be a non-trivial free product with amalgamation. , S; rels S,t~XSltl = fl(S1), .... whose base S is a tree product . Each vertex group in the base is a conjugate of G~ A H or G2 N H and each amalgamated subgroup is a conjugate of A intersected with H. Further the associated subgroups {S~,f~(Si)}are also conjugates of A intersected wi~h H and each associated subgroup is contained in a vertex group. An HNNgroup whose base is a tree product and where each associated subgroup is a subgroup of a vertex group is called a treed HNNgroup.

Then there exists a tree such that G acts on X with fundamental domain a segment P ~ Q, and such that ~p = G1, GQ, -~- ~2 and Gy =- A for the respective stabilizers. This theorem establishes an equivalence between free products with amalgamation and groups acting on trees with a segment as a fundamental domain. The next result establishes a similar equivalence between HNN groups and group actions on trees with loops as the quotient graph. 3. (1) Let G act on tree X with a loop {as above} as the quotient graph for X rood G.

In general, as already mentioned, for a given finite system, a suitable order can always be chosen such that this finite system can be carried by a Nielsen transformation into a minimal system. The Nielsen reduction method in G now refers to Nielsen transformations from given systems to shorter systems and the resulting investigation of minimal systems. An analysis of the result of H. Zieschang [Z 2] for G {see also Rosenberger [R 6]} produces the following result. 2. Let G = HI *A H2. , y,~} for which one of ~he following cases hold: (i) y~ = for so me i ~ {1, ..

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