Virtually free groups and integral representations
A virtually free group \(G\) is an extension of a (non abelian) free group \(F\) by a finite group \(H\). The action of \(H\) on \(F\) induces in a natural way the action of \(H\) on the abelization \(F^{\mathrm{ab}}=F/F'\) of \(F\) that turns \(F^{\mathrm{ab}}\) into a \(\mathbb{Z}[H]\)-module. The objective of this paper is to clarify some issues such as the structure of such a module, if, and in which cases, it is decomposable and determining its indecomposable components. By results of \textit{A. Karrass} et al. [J. Aust. Math. Soc. 16, 458--466 (1973; Zbl 0299.20024)], \textit{D. E. Cohen} [``Groups with free subgroups of finite index'', Lect. Notes Math. 319, 26--44 (1973; \url{doi:10.1007/BFb0058927})] and Scott \(G =\pi_{1}(\mathcal{G},\Gamma)\) is the fundamental group of a graph of finite groups. Then the authors describe the structure of \(\mathbb{Z}[H]\)-module \(F^{\mathrm{ab}}\) in terms of the graph of the group \(\pi_{1}(\mathcal{G},\Gamma)\). The main result (Theorem 1.1) is too technical to be stated here; an interesting consequence is the following corollary: Theorem 1.2. Let \(H\) be a finite \(p\)-group. Then \(F^{\mathrm{ab}}\) is a permutation \(\mathbb{Z}[H]\)-module if and only if \(G\) is an \(HNN\)-extension of \(H\) (with finitely many stable letters). Some interesting examples are also provided which illustrate the achieved results.
Bass-Serre theory of groups acting on trees
scientific article; zbMATH DE number 6845383
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