Two generator subgroups of free products with commuting subgroups
2-generator subgroups of products with commuting subgroups are studied. The concept of a free product with commuting subgroups is a natural intermediate construction between a free product (the commuting subgroups are trivial) and a direct product (the commuting subgroups equal the factors), just like a graph group lives between a free and a free Abelian group. Automorphisms of graph groups were studied and, using geometric techniques, the centralizer problem was solved by \textit{H. Servatius} [J. Algebra 126, No. 1, 34-60 (1989; Zbl 0682.20022)]. Here the geometric techniques developed by \textit{I. Kapovich} and \textit{R. Weidmann} [Arch. Math 73, No. 3, 172-181 (1999; Zbl 0988.20012)] are used to show that a 2-generator subgroup of a product of free groups with commuting subgroups is either free, or of the form \(\langle a,b\mid[a^p,b^q]\rangle\) for some natural exponents \(p\) and \(q\), or it is the fundamental group of a graph of cyclic groups were the underlying graph is either homeomorphic to a segment or a circle.
scientific article; zbMATH DE number 1787470
scientific article