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 | References | Similar Articles | Additional Information

Abstract: Let be a finitely generated residually finite group and let denote the number of index subgroups of . If for some and for all , then is said to have polynomial subgroup growth (PSG, for short). The degree of is then defined by .

Very little seems to be known about the relation between and the algebraic structure of . 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 is a finite index subgroup, then .

A large part of the paper is devoted to the structure of groups of small degree. We show that is bounded above by a linear function of if and only if is virtually cyclic. We then determine all groups of degree less than , 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 .

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

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02220-5

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