TY - JOUR

T1 - On the degree of groups of polynomial subgroup growth

AU - Shalev, Aner

PY - 1999

Y1 - 1999

N2 - Let G be a finitely generated residually finite group and let an(G) denote the number of index n subgroups of G. If an(G) ≤ na for some a and for all n, then G is said to have polynomial subgroup growth (PSG, for short). The degree of G is then defined by deg(G) = lim sup log an(G)/log n Very little seems to be known about the relation between 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 ≤ G is a finite index subgroup, then deg(G) ≤ deg(H) + 1. A large part of the paper is devoted to the structure of groups of small degree. We show that a(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 àla Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.

AB - Let G be a finitely generated residually finite group and let an(G) denote the number of index n subgroups of G. If an(G) ≤ na for some a and for all n, then G is said to have polynomial subgroup growth (PSG, for short). The degree of G is then defined by deg(G) = lim sup log an(G)/log n Very little seems to be known about the relation between 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 ≤ G is a finite index subgroup, then deg(G) ≤ deg(H) + 1. A large part of the paper is devoted to the structure of groups of small degree. We show that a(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 àla Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.

UR - http://www.scopus.com/inward/record.url?scp=22844455923&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-99-02220-5

DO - 10.1090/s0002-9947-99-02220-5

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AN - SCOPUS:22844455923

SN - 0002-9947

VL - 351

SP - 3793

EP - 3822

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 9

ER -