A quite useful philosophy in mathematics is to use the sharpness of an inequality regarding the "shape" of a topological space in order to detect a precise geometry on it: more precisely, the maximal value of the inequality usually allows to identify a specific geometric structure. Think for instance either to the applications of arithmetic/geometric mean inequality or to the isoperimetric inequality on the plane. Something similar happens in the world of Zimmer's cocycle theory. In this seminar we are going to focus our attention on Zimmer's cocycles associated to the fundamental group a surface S with genus bigger than or equal to 2. If such a measurable cocycle admits a (generalized) boundary map, one can define the notion of Euler number. The latter well behaves along cohomology classes and its absolute value is bounded by the modulus of the Euler characteristic of S. Remarkably the maximal value is attained if and only if the cocycle is cohomologous to a hyperbolization. The first part of the talk will be a gentle introduction to measurable cocycles and boundary theory. Then, we are going to introduce the orientation cocycle on the circle. Finally we will define the Euler number of a measurable cocycle and we will discuss its rigidity property. This is a joint work with Marco Moraschini.