On groups of polynomial subgroup growth

Alexander Lubotzky*, Avinoam Mann

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

44 Scopus citations

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 languageEnglish
Pages (from-to)521-533
Number of pages13
JournalInventiones Mathematicae
Volume104
Issue number1
DOIs
StatePublished - Dec 1991

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