The asymptotic distribution of lattice points in hyperbolic space

Suppose x and y are two points in the upper half-plane H+, and suppose Γ is a discontinuous group of conformal automorphisms of H+ having compact fundamental domain S. Denote by NT(x, y) the number of points of the form γy (γ ϵ Γ) in the closed disc of hyperbolic radius T centered about x, and set QT(x, y) = NT(x, y) − V(T)A, where V(T) is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ⊢LxL(QT(x,y))2 is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.