Benjamini-Schramm convergence of normalized characteristic numbers

On a class of closed oriented Riemannian manifolds with lower bounds on Ricci curvature and injectivity radius we study the parameter given by a characteristic number over the volume of the manifold. The domain of this parameter is endowed with the Benjamini-Schramm (BS) topology. The BS-topology is a weak notion of convergence, that originated from graph theory, where it is insensively studied, but also has applications in various other areas. Using an integral characterization (actually, a generalization of Gauss-Bonnet formula) one can show that the parameter explained is continuous and has a continuous extension to the completion of its domain in the BS topology. From the known fact that this completion is compact one can derive that the parameter is testable in constant time.