Valuations, completions, and hyperbolic actions of metabelian groups (Q6561000)

The main theorems of the paper under review (Theorems 1.1 and 1.2) reduce the problem of classifying hyperbolic structures of \textit{abelian-by-cyclic} groups (that is, groups isomorphic to a semidirect product of the form \(A\rtimes \mathbb Z\) for some finitely generated, torsion-free abelian group~\(A\)) based on admissible matrices \(\gamma\) to the following ones: computing the invariant subspaces of~\(\gamma\) and computing the prime factorization of the characteristic polynomial of \(\gamma\) in the formal power series ring~\(\mathbb Z[[x]]\). The former is straightforward using Jordan normal forms, while the latter can be algorithmically solved. The authors show how to use the main results of the paper to classify the hyperbolic actions of a number of groups, among which the lamplighter groups (Section 5.1) and the solvable Baumslag-Solitar groups (Section 5.2.1).\N\NRecall that for any abelian-by-cyclic group \(G\), there is an endomorphism \(\gamma_G\in\operatorname{M}_n(\mathbb Z)\) of \(\mathbb Z^n\) with \(\operatorname{det}(\gamma_G)\neq0\) and \(G=\langle \mathbb Z^n,t\,:\, tzt^{-1}=\gamma_G(z),\,\forall z\in\mathbb Z^n\rangle\).