Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [B] A. Babakhanian, Cohomological Methods in Group Theory, Dekker, New York, 1972. MR 41:6977
  • [Ba] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. 25 (1972), 603-614. MR 52:577
  • [CM] H.S.M. Coxeter and W.O.J. Moser, Generators and Relators for Discrete Groups, Springer, Berlin, 1957. MR 19:527d
  • [dS] M.P.F. du Sautoy, Finitely generated groups, $p$-adic analytic groups and Poincaré series, Ann. of Math. 137 (1993), 639-670. MR 94j:20029
  • [E] W. Ellison and F. Ellison, Prime Numbers, Wiley, New York, 1985. MR 87a:11082
  • [Gr] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53 (1981), 53-78. MR 83b:53041
  • [GSS] F.J. Grunewald, D. Segal and G.C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), 185-223. MR 89m:11084
  • [HKLSh] E. Hrushovski, P.H. Kropholler, A. Lubotzky and A. Shalev, Powers in finitely generated groups, Trans. Amer. Math. Soc. 348 (1996), 291-304. MR 96f:20061
  • [HW] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (3rd edition), Clarendon, Oxford, 1954. MR 16:673c
  • [L1] A. Lubotzky, Subgroup growth and congruence subgroups, Invent. Math. 119 (1995), 267-295. MR 95m:20054
  • [L2] A. Lubotzky, Counting finite index subgroups, Groups '93 - Galway / St Andrews, London Math. Soc. Lecture Note Series 212, Cambridge University Press, Cambridge, 1995, pp. 368-404. MR 96h:20080
  • [L3] A. Lubotzky, Subgroup growth, Proc. ICM Zürich '94, vol. 1, Birkhäuser, Basel, 1995, 309-317. MR 97k:20048
  • [LM] A. Lubotzky and A. Mann, On groups of polynomial subgroup growth, Invent. Math. 104 (1991), 521-533. MR 92a:20038
  • [LMS] A. Lubotzky, A. Mann and D. Segal, Finitely generated groups of polynomial subgroup growth, Israel. J. Math. 82 (1993), 363-371. MR 95b:20051
  • [M] A. Mann, Some properties of polynomial subgroup growth groups, Israel J. Math. 82 (1993), 373-380. MR 94m:20088
  • [MS1] A. Mann and D. Segal, Uniform finiteness conditions in residually finite groups, Proc. London Math. Soc. (3) 61 (1990), 529-545. MR 91j:20093
  • [MS2] A. Mann and D. Segal, Subgroup growth: survey of current results, Infinite Groups 94, ed. de Giovanni and Newell, Walter de Gruyter, Berlin-New York, 1995, pp. 179-197. CMP 98:03
  • [Me] A.D. Mednykh, On the number of subgroups in the fundamental group of a closed surface, Comm. Alg. 16(10) (1988), 2137-2148. MR 90a:20076
  • [N] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warsaw, 1974. MR 50:268
  • [R1] D.J.S. Robinson, Finiteness Conditions and Generalized Soluble Groups, I-II, Springer, New York, 1972. MR 48:11314; MR 48:11315
  • [R2] D.J.S. Robinson, Splitting theorems for infinite groups, Symp. Math. XVII (1976), 441-470. MR 53:10936
  • [R3] D.J.S. Robinson, A Course in the Theory of Groups, Springer, New York, 1982. MR 84k:20001
  • [SSh1] D. Segal and A. Shalev, Groups of fractionally exponential subgroup growth, J. Pure and Appl. Alg. 88 (1993), 205-223. MR 94e:20047
  • [SSh2] D. Segal and A. Shalev, Profinite groups with polynomial subgroup growth, J. London Math. Soc. 55 (1997), 320-334 (Hartley memorial issue). MR 98c:20053
  • [Sh1] A. Shalev, Growth functions, $p$-adic analytic groups, and groups of finite coclass, J. London Math. Soc. (2) 46 (1992), 111-122. MR 94a:20047
  • [Sh2] A. Shalev, Subgroup growth and sieve methods, Proc. London Math. Soc. 74 (1997), 335-359. MR 98c:20054
  • [Sh3] A. Shalev, Groups whose subgroup growth is less than linear, Int. J. Alg. and Comp. 7 (1997), 77-91. MR 98g:20046
  • [S] G.C. Smith, Zeta-Functions of Torsion-Free Finitely Generated Nilpotent Groups, Ph.D. Thesis, UMIST, Manchester, 1983.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20E07, 20E34

Retrieve articles in all journals with MSC (1991): 20E07, 20E34

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

American Mathematical Society