The free product of groups with amalgamated subgroup malnormal in a single factor.

*(English)*Zbl 0931.20022A proper subgroup \(S\) of \(A\) is said to be malnormal in \(A\) if the intersection of the subgroups \(aSa^{-1}\) and \(S\) is trivial if and only if \(a\) is an element of \(A\) outside of \(S\). The authors discuss groups that are free products with amalgamation, where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. When the amalgamating subgroup is malnormal in a single factor and the global group is of rank two, the authors show that either the non-malnormal factor contains a torsion element or, if not, then there is a generating pair of one of four specific types. For each type, the authors establish a set of relations which must hold in the factor \(B\) and give restrictions on the rank of generators of each factor.

Reviewer: Alexandr Ivanov Budkin (Barnaul)

##### MSC:

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20F05 | Generators, relations, and presentations of groups |

20E34 | General structure theorems for groups |

##### Keywords:

malnormal subgroups; free products with amalgamation; generating pairs; relations; generators
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\textit{S. A. Bleiler} and \textit{A. C. Jones}, J. Pure Appl. Algebra 127, No. 2, 119--136 (1998; Zbl 0931.20022)

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