Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍⁿ. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.

As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.

In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

以下最大表示理论背后的理念，我们引入一个Zimmer的闭链Γ×的体积X →PO°（N， 1），其中Γ是一个免费的扭转（非）均匀PO晶格°（Ñ，1 ），其中n > 3，并且X是合适的标准Borel概率Γ空间。我们的数值不变式将（非）均匀晶格的表示量扩展到了可测量的cocycles，并且在统一设置中，它与自耦合欧拉数的广义形式一致。我们证明了我们在歧管Γ\ℍ的体积而言余循环满足一个米尔诺木型不等式的体积^ñ。另外，该不变量可以解释为有界同调类之间的合适的乘法常数。这使我们可以定义一个数量逐渐消失的可测量的cocycle。相同的解释使我们能够通过具有基本恒定符号的可测图X →PO°（n， 1）来表征与标准晶格嵌入所诱导的同循环同调的最大同循环。

作为我们对cocycles数量的刚性结果的副产品，我们给出了映射度定理的另一种证明。这使我们能够以最大的cocycles完整描述封闭双曲线流形之间​​同位到局部等位图。

在维度n中= 2，我们引入了与一个封闭曲面组相关的可测量Cocycle的Euler数的概念，并证明了它扩展了经典的Euler表示形式。我们的欧拉数与自耦联的欧拉数的广义形式一致，直到乘法常数。模仿在体积情况下开发的技术，我们显示了Milnor-Wood型不等式，其上限由相关封闭表面的欧拉特性模量给出。对于欧拉数自耦联的广义版本，这提供了相同结果的替代证明。最后，使用欧拉数作为有界同性类之间的乘性常数的解释，我们将最大的同化环表征为与超代谢诱导的同化环同系。