The free product of groups with amalgamated subgroup malnormal in a single factor. (English) Zbl 0931.20022

A 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.


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
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[1] Baumslag, B., Generalized free products whose two-generator subgroups are free, J. London Math. Soc., 43, 601-606 (1968) · Zbl 0172.02702
[2] Bleiler, S.; Jones, A. C., On two generator satellite knots, MSRI preprint 1997-023 (1997) · Zbl 1053.57001
[3] Jones, A. C., Composite two-generator links have Hopf link summand (1993), preprint
[4] Karrass, A.; Solitar, D., The free product of two groups with a malnormal amalgamated subgroup, Can. J. Math., 23, 6, 933-959 (1971) · Zbl 0247.20028
[5] Kirby, R., Problems in low-dimensional topology, (Proc. Symp. Pure Math., 32 (1978)), 273-312
[6] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial Group Theory (1966), Interscience Publishers: Interscience Publishers New York
[8] Norwood, F. H., Every two-generator knot is prime, (Proc. Am. Math. Soc., 86 (1982)), 143-147 · Zbl 0506.57004
[9] Stallings, J., A topological proof of Grushko’s theorem on free products, Math. Z., 90, 1-8 (1965) · Zbl 0135.04603
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