In this article we discuss counting and equidistribution problems for orbits of thin groups in homogeneous spaces.

Let G be a connected semisimple Lie group and H a closed subgroup. We consider the homogeneous space V = H\G and fix the identity coset x0 = [e]. Let Γ be a discrete subgroup of G such that the orbit x0Γ is discrete, and let {BT : T >1} be a family of compact subsets of V whose volume tends to infinity as T →∞. Understanding the asymptotic of #(x0 ∩ BT ) is a fundamental problem which bears many applications in number theory and geometry. We refer to this type of counting problem as an archimedean counting as opposed to a combinatorial counting where the elements in x0 are ordered with respect to a word metric on Γ; both have been used in applications to sieves; see [Kowalski 2010; 2014].