On groups of polynomial subgroup growth

Let Γ be a finitely generated group and a_n(Γ)=the number of its subgroups of index n. We prove that, assuming Γ is residually nilpotent (e.g., Γ linear), then a_n(Γ) grows polynomially if and only if Γ is solvable of finite rank. This answers a question of Segal. The proof uses a new characterization of p-adic analytic groups, the theory of algebraic groups and the Prime Number Theorem. The method can be applied also to groups of polynomial word growth.