By the Gauss-Bonnet theorem, the volume of a hyperbolic 4-manifold is proportional to its Euler characteristic. There are examples, constructed explicitly in Ratcliffe, John G.; Tschantz, Steven T. The volume spectrum of hyperbolic 4-manifolds. Experiment. Math. 9 (2000), no. 1, 101—125. of hyperbolic 4-manifolds with every positive integer as their […]