A lattice problem for differential forms in Euclidean spaces (Q909709)

Let \(\Gamma\) be a properly discontinuous group of isometries of n- dimensional Euclidean space acting cocompactly. Introduce the counting function \[ A_{\Gamma}(t,x,y)=card\{\gamma \in \Gamma | \quad d(x,y)<t\}. \] There is a wealth of results on the asymptotic behaviour of \(A_{\Gamma}\) as \(t\to \infty\). The author studies a weighted version of the functions \(A_{\Gamma}\). The weights coming from translation length of the \(\gamma\) on differential forms. The author gives a principal term and an O-term for his counting functions. The interest lies here in the strong O-term. The principal term can (asymptotically) be found by considering Epstein-\(\zeta\)-functions with spherical functions as coefficients and using Tauberian theory.