Finitely presented subgroups of a product of two free groups (Q2716952)

The paper gives a new proof for the following theorem of \textit{G. Baumslag} and \textit{J. E. Roseblade}: let \(H\) be a finitely presented subgroup of a direct product \(A\times B\) of two free groups \(A\) and \(B\). If \(H\) is not free, then \(H\) contains a subgroup \(A'\times B'\) of finite index, where \(A'\) (respectively \(B'\)) is a subgroup of \(A\) (respectively of \(B\)) [J. Lond. Math. Soc., II. Ser. 30, 44-52 (1984; Zbl 0559.20018)].NEWLINENEWLINENEWLINEThe new proof offered is based on the combinatorial geometric method of the van Kampen diagrams [for background information on the topic see \textit{R. C. Lyndon} and \textit{P. E. Schupp}, Combinatorial group theory (1977; Zbl 0368.20023) or \textit{A. Yu. Ol'shanskij}, Geometry of defining relations in groups (1989; Zbl 0676.20014)]. The original proof of this theorem is based on group homology and special sequences. There is another geometric proof of the theorem considered given by \textit{M. R. Bridson} and \textit{D. T. Wise} and based on \({\mathcal V}H\)-complexes [Math. Proc. Camb. Philos. Soc. 126, No. 3, 481-497 (1999; Zbl 0942.20009)]. The paper reviewed also contains a brief outline of the van Kampen diagrams and two examples showing that the parts ``finitely presented'' of the theorem and ``finite index'' are necessary.