Group boundaries for semidirect products with \(\mathbb{Z}\) (Q6594552)

The notion of \(\mathcal{Z}\)-group was introduced by \textit{M. Bestvina} in [Mich. Math. J. 43, No. 1, 123--139 (1996; Zbl 0872.57005)], in order to unify the study of Gromov boundaries of hyperbolic groups with visual boundaries of CAT(0) groups and to provide a framework for assigning boundaries to a wider class of groups.\N\NThe first three main results in this paper under review are the following.\N\NTheorem 1.1: If a torsion-free group \(G\) admits a \(\mathcal{Z}\)-structure with boundary \(Z\), then every semidirect product \(G \rtimes \mathbb{Z}\) admits a \(\mathcal{Z}\)-structure with boundary equal to the suspension of \(Z\).\N\NTheorem 1.2: Every closed \(3\)-manifold group admits a \(\mathcal{Z}\)-structure.\N\NTheorem 1.3: Every strongly polycyclic and every finitely generated nilpotent group \(G\) admits a \(\mathcal{Z}\)-structure with a \(k\)-sphere as boundary for some \(k \geq -1\).\N\NThe concept E\(\mathcal{Z}\)-structure, a refinement of \(\mathcal{Z}\)-structures, was defined by \textit{F. T. Farrell} and \textit{J.-F. Lafont} in [Comment. Math. Helv. 80, No. 1, 103--121 (2005; Zbl 1094.57003)] and has proven useful in attacks on the Novikov conjecture and related problems. The following interesting theorem (Theorem 1.6) is a consequence of a more general result demonstrated in the paper: Suppose \(G\) is a hyperbolic group, a finitely generated abelian group, or a CAT(0) group with the isolated flats property. Then, for any \(\varphi \in \mathrm{Aut}(G)\), \(G \rtimes \mathbb{Z}\) admits an E\(\mathcal{Z}\)-structure with boundary equal to the suspension of the Gromov or visual boundary of \(G\).\N\NBy the results of Farrell and Lafont ([loc. cit.]), Theorem 1.6 implies the Novikov conjecture, for the groups covered there, whenever they are torsion-free (other proofs of Novikov conjecture are known in these particular cases).