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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the degree of groups
of polynomial subgroup growth

Author: Aner Shalev
Journal: Trans. Amer. Math. Soc. 351 (1999), 3793-3822
MSC (1991): Primary 20E07, 20E34
Published electronically: April 20, 1999
MathSciNet review: 1475693
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Abstract: Let $G$ be a finitely generated residually finite group and let $a_n(G)$ denote the number of index $n$ subgroups of $G$. If $a_n(G) \le n^{\alpha}$ for some $\alpha$ and for all $n$, then $G$ is said to have polynomial subgroup growth (PSG, for short). The degree of $G$ is then defined by ${\mathrm{deg}}(G) = \limsup {{\log a_n(G)} \over {\log n}}$.

Very little seems to be known about the relation between ${\mathrm{deg}}(G)$ and the algebraic structure of $G$. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if $H \le G$ is a finite index subgroup, then ${\mathrm{deg}}(G) \le {\mathrm{deg}}(H)+1$.

A large part of the paper is devoted to the structure of groups of small degree. We show that $a_n(G)$ is bounded above by a linear function of $n$ if and only if $G$ is virtually cyclic. We then determine all groups of degree less than $3/2$, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval $(1, 3/2)$.

Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.

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Additional Information

Aner Shalev
Affiliation: Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

Received by editor(s): July 10, 1996
Received by editor(s) in revised form: March 8, 1997
Published electronically: April 20, 1999
Additional Notes: This work was supported in part by a grant from the Israel Science Foundation
Article copyright: © Copyright 1999 American Mathematical Society