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.

中文翻译：

---

圆柱双曲群有界上同调的零范数子空间

我们构造了在圆上纤维化的双曲三流形的组合体积形式。这些形式定义了有界上同调中的非平凡类。在引入精确有界上同调的新半范数后，我们使用这些组合类来证明，在 3 次中，圆柱双曲群的有界上同调的零范数子空间是无限维的。在附录中，我们使用相同的技术给出了下界的上同调证明，最初是由于 Brock，在封闭曲面的共界伪 Anosov 同胚的映射环的体积上，根据其 Teichm\"uller 平移距离。