On the degree of groups of polynomial subgroup growth
Author:
Aner Shalev
Journal:
Trans. Amer. Math. Soc. 351 (1999), 37933822
MSC (1991):
Primary 20E07, 20E34
Published electronically:
April 20, 1999
MathSciNet review:
1475693
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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 numbertheoretic, 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/S0002994799022205
PII:
S 00029947(99)022205
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
