Hyperbolic groups containing subgroups of type \(\mathscr{F}_3\) not \(\mathscr{F}_4\) (Q6593654)

The first example of a subgroup of a hyperbolic group which is finitely presented but not of type \(\mathscr{F}_{3}\), and thus not itself hyperbolic, was constructed by \textit{N. Brady} [J. Lond. Math. Soc., II. Ser. 60, No. 2, 461--480 (1999; Zbl 0940.20048)]. \textit{G. Italiano} et al. [Invent. Math. 231, No. 1, 1--38 (2023; Zbl 1512.57039)], constructed the first examples of subgroups of hyperbolic groups which are not hyperbolic but have finite classifying spaces.\N\NThe main result in the paper under review is Theorem 1: There exist infinitely many pairwise non-isomorphic hyperbolic groups \(G\) admitting a surjective homomorphism \(\phi: G \rightarrow \mathbb{Z}\) whose kernel is of type \(\mathscr{F}_{3}\) and has the property that \(H_{4}(\ker(\phi), \mathbb{Z})\) is not finitely generated. In particular, \(\ker(\phi)\) is not of type \(\mathscr{F}_{4}\).\N\NThese groups are obtained by Dehn filling starting from a non-uniform lattice in \(\mathrm{PO}(8,1)\).