  We construct combinatorial volume forms of hyperbolic three manifolds
fibering over the circle. These forms define non-trivial classes in bounded
cohomology. After introducing a new seminorm on exact bounded cohomology, we
use these combinatorial classes to show that, in degree 3, the zero norm
subspace of the bounded cohomology of an acylindrically hyperbolic group is
infinite dimensional. In the appendix we use the same techniques to give a
cohomological proof of a lower bound, originally due to Brock, on the volume of
the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed
surface in terms of its Teichm\"uller translation distance.
