The hyperbolic lattice point count in infinite volume with applications to sieves

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms that are uniform as the lattice moves through " congruence" subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f (c, d) = c² + d², but restrict (c, d) to the orbit O{script} = (0, 1)Γ,whereΓ is an infinite-index, nonelementary, finitely generated subgroup of SL(2, Z{double-struck}). Assume that the Reimann surface Γ\H{double-struck} has a cusp at infinity. We show that the set of values f (O{script}) contains infinitely many integers having at most R prime factors for any R>4/(δ - Θ),whereΘ>1/2 is the spectral gap and δ<1 is the Hausdorff dimension of the limit set of Γ.Ifδ>149/150, then we can take Θ = 5/6, giving R = 25. The limit of this method is R = 9 for δ - Θ>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.