  Poisson processes in the space of $(d-1)$-dimensional totally geodesic
subspaces (hyperplanes) in a $d$-dimensional hyperbolic space of constant
curvature $-1$ are studied. The $k$-dimensional Hausdorff measure of their
$k$-skeleton is considered. Explicit formulas for first- and second-order
quantities restricted to bounded observation windows are obtained. The central
limit problem for the $k$-dimensional Hausdorff measure of the $k$-skeleton is
approached in two different set-ups: (i) for a fixed window and growing
intensities, and (ii) for fixed intensity and growing spherical windows. While
in case (i) the central limit theorem is valid for all $d\geq 2$, it is shown
that in case (ii) the central limit theorem holds for $d\in\{2,3\}$ and fails
if $d\geq 4$ and $k=d-1$ or if $d\geq 7$ and for general $k$. Also rates of
convergence are studied and multivariate central limit theorems are obtained.
Moreover, the situation in which the intensity and the spherical window are
growing simultaneously is discussed. In the background are the Malliavin-Stein
method for normal approximation and the combinatorial moment structure of
Poisson U-statistics as well as tools from hyperbolic integral geometry.
