Hyperbolic manifolds that fibre algebraically up to dimension 8

We construct some cusped finite-volume hyperbolic n-manifolds Mn that fibre algebraically in all the dimensions 5≤n≤8. That is, there is a surjective homomorphism π1(Mn)→Z with finitely generated kernel. The kernel is also finitely presented in the dimensions n=7,8, and this leads to the first examples of hyperbolic n-manifolds M˜n whose fundamental group is finitely presented but not of finite type. These n-manifolds M˜n have infinitely many cusps of maximal rank and, hence, infinite Betti number bn−1. They cover the finite-volume manifold Mn. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes P5,…,P8, and then applying some arguments of Jankiewicz, Norin and Wise [18] and Bestvina and Brady [7]. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to Pn, and the algebra of integral octonions for the crucial dimensions n=7,8.