Finite presentation of fibre products of metabelian groups
Associated to each pair of short exact sequences of groups \(1\to N_i\to\Gamma_i@>p_i>>Q\to 1\), \(i=1,2\), one has the fibre product \(P=\{(\gamma_1,\gamma_2)\in\Gamma_1\times\Gamma_2\mid p_1(\gamma_1)=p_2(\gamma_2)\}\). The paper under review concerns entirely the case \(\Gamma_1=\Gamma_2=\Gamma\), \(N_1=N_2=N\) and for the most part it is focused on the case where \(p_1=p_2\), where in this case the fibre product is called `untwisted'. The main problem of the paper under review is when such fibre products are finitely presented. There are several results in this direction. If \(\Gamma\) is free or a surface group and both \(Q\) and \(N\) are infinite, then \(P\) is never finitely presented [see \textit{G. Baumslag, J. E. Roseblade}, J. Lond. Math. Soc., II. Ser. 30, 44-52 (1984; Zbl 0559.20018), \textit{F. J. Grunewald}, J. Lond. Math. Soc., II. Ser. 17, 427-436 (1978; Zbl 0385.20020) and \textit{M. R. Bridson, J. Howie, C. F. Miller, III} and \textit{H. Short}, Geom. Dedicata 92, 95-103 (2002; Zbl 1048.20009)]. Here the authors prove in Theorem 1 that if \(\Gamma\) is a finitely presented metabelian group, then the untwisted fibre product associated to any short exact sequence \(1\to N\to\Gamma\to Q\to 1\) is finitely presented. Theorem 9 of the paper under review is about a certain twisted case as follows: Let \(1\to N\to\Gamma@>p>>Q\to 1\) be a short exact sequence with \(Q\) and \(N\) Abelian. Consider the twisted fibre product \(P=\{(\gamma,\gamma')\mid p(\gamma)=-p(\gamma')\}\) associated to this sequence. Then \(P\) is finitely presented if and only if \(\Gamma\) is polycyclic. The main tools for the proof of Theorem 1 and Theorem 9 are a fundamental theorem of \textit{R. Bieri} and \textit{R. Strebel} which gives a necessary and sufficient condition for \(\Gamma\) to be finitely presented and also their characterization of when \(\Gamma\) is polycyclic [see Proc. Lond. Math. Soc., III. Ser. 41, 439-464 (1980; Zbl 0448.20029)]. A number of examples are also constructed, including a non-finitely presented metabelian group \(P\) with \(H_2(P,\mathbb{Z})\) finitely generated.
scientific article; zbMATH DE number 1915688
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