Abstract
Let Γ be a finitely generated group and an(Γ)=the number of its subgroups of index n. We prove that, assuming Γ is residually nilpotent (e.g., Γ linear), then an(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.
Original language | English |
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Pages (from-to) | 521-533 |
Number of pages | 13 |
Journal | Inventiones Mathematicae |
Volume | 104 |
Issue number | 1 |
DOIs | |
State | Published - Dec 1991 |