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 N_T(x, y) the number of points of the form γy (γ?Γ) in the closed disc of hyperbolic radius T centered about x, and set $\text{Q}{}_{\mathrm{T}}\text{(x, y) = N}{}_{\mathrm{T}}\text{(x, y) ?}\text{V(T)}\text{A}\text{,}\text{where}\text{V(T)}$ is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ?_LxL(Q_T(x,y))² is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.