  We study topological and geometric functionals of $l_\infty$-random geometric
graphs on the high-dimensional torus in a sparse regime, where the expected
number of neighbors decays exponentially in the dimension. More precisely, we
establish moment asymptotics, functional central limit theorems and Poisson
approximation theorems for certain functionals that are additive under disjoint
unions of graphs. For instance, this includes simplex counts and Betti numbers
of the Rips complex, as well as general subgraph counts of the random geometric
graph. We also present multi-additive extensions that cover the case of
persistent Betti numbers of the Rips complex.
