Counting arithmetic lattices and surfaces

We give estimates on the number AL_H(x) of conjugacy classes of arithmetic lattices Γ of covolume at most x in a simple Lie group H. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most x. Our main result is for the classical case H=PSL(2;R{double-struck}) where we show that The proofs use several different techniques: geometric (bounding the number of generators of Γ as a function of its covolume), number theoretic (bounding the number of maximal such Γ) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of Γ).