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

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

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.

**[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, -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, -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.

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Additional Information

**Aner Shalev**

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

DOI:
https://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