### Matrix coefficients, counting and primes for orbits of geometrically finite groups

Let $G:=\mathrm{SO}{(n,1)}^{\circ}$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma $ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{\mathcal{B}}_{T}\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {\mathcal{B}}_{T}=\mathcal{M}\left({\mathcal{B}}_{T}\right)+O\left(\mathcal{M}{\left({\mathcal{B}}_{T}\right)}^{1-\eta}\right)$ for an explicit measure $\mathcal{M}$ on $H\setminus G$ which depends on $\Gamma $. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma $ in arithmetic settings....