Poisson hyperplanes in hyperbolic space
Abstract: In the focus of this talk are random tessellations in hyperbolic space induced by Poisson point processes on the space of hyperbolic hyperplanes (totally geodesic subspaces of co-dimension 1). In the first part of the talk we consider the so-called k-skeleton of such tessellations and prove that, when observed in a sequence of increasing observation windows, the k-volume of the k-skeleton satisfies a central limit theorem only for dimensions 2 and 3 and that asymptotic normality fails in all higher dimensions. We indicate possible generalizations to Poisson processes of lower-dimensional random subspaces as well. If time permits we also describe a way to address the combinatorial structure of the zero cell of a hyperbolic hyperplane tessellation. In particular we present a fully explicit formula for the number of facets of this cell.