Counting arithmetic lattices and surfaces

الأصل · EN

We give estimates on the number ALₕ(x) 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) where we compute the limit of ALₕ(x) / x x when x→∞. 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).