  We present new Poisson process approximation results for stabilizing
functionals of Poisson and binomial point processes. These functionals are
allowed to have an unbounded range of interaction and encompass many examples
in stochastic geometry. Our bounds are derived for the Kantorovich-Rubinstein
distance using the generator approach to Stein's method. We give different
types of bounds for different point processes. While some of our bounds are
given in terms of coupling of the point process with its Palm version, the
others are in terms of the local dependence structure formalized via the notion
of stabilization. We provide two supporting examples for our new framework -
one is for Morse critical points of the distance function, and the other is for
large k-nearest neighbor balls. Our bounds considerably extend the results in
Barbour and Brown (1992), Decreusefond, Schulte and Thale (2016) and Otto
(2020).
