Homology Growth, Hyperbolization, and Fibering

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and—for almost all primes p — F ₚ -homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic 7-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.