Summary
Let Γ be a finitely generated group anda n (Γ)=the number of its subgroups of indexn. We prove that, assuming Γ is residually nilpotent (e.g., Γ linear), thena 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 ofp-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.
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Oblatum 1-VII-1989 & 7-VI-1990
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Lubotzky, A., Mann, A. On groups of polynomial subgroup growth. Invent. math. 104, 521–533 (1991). https://doi.org/10.1007/BF01245088
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DOI: https://doi.org/10.1007/BF01245088