HYPERBOLIC MANIFOLDS THAT FIBRE ALGEBRAICALLY UP TO DIMENSION 8

We construct some cusped finite-volume hyperbolic n-manifolds M-n that fibre algebraically in all the dimensions 5 <= n <= 8. That is, there is a surjective homomorphism pi(1)(M-n) -> 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) over tilde (n) whose fundamental group is finitely presented but not of finite type. These n-manifolds (M) over tilde (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, ..., P-8, 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 Cosset polytopes dual to P-n, and the algebra of integral octonions for the crucial dimensions n = 7,8.