  We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that
fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a
surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ 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 $\widetilde M_n$ whose
fundamental group is finitely presented but not of finite type. These
$n$-manifolds $\widetilde M_n$ have infinitely many cusps of maximal rank and
hence infinite Betti number $b_{n-1}$. They cover the finite-volume manifold
$M_n$.
  We obtain these examples by assigning some appropriate colours and states to
a family of right-angled hyperbolic polytopes $P_5, \ldots, P_8$, and then
applying some arguments of Jankiewicz, Norin, Wise and Bestvina, Brady. We
exploit in an essential way the remarkable properties of the Gosset polytopes
dual to $P_n$, and the algebra of integral octonions for the crucial dimensions
$n=7,8$.
