Growth functions, p-adic analytic groups, and groups of finite coclass

Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


Certain growth functions associated with a pro-p group G are studied, and their impact on the structure of G is described. In particular we prove that (1) if, for some c < § and for all sufficiently large k, G has at most open subgroups of index pk, then G is p-adic analytic; (2) if, for some k, then G is p-acid analytic; (3) if, for some n, (G:Gn) <pn+[l0gP(n/2), where Gn is the lower central series of G, then G is abelian-by-finite. Results (1) to (3) sharpen previous results of Lubotzky and Mann, Lazard, Donkin and Leedham-Green respectively. Related constructions, showing that some of these results are asymptotically best possible, are analysed in some detail.

Original languageAmerican English
Pages (from-to)111-122
Number of pages12
JournalJournal of the London Mathematical Society
Issue number1
StatePublished - Aug 1992
Externally publishedYes

Bibliographical note

Funding Information:
Received 22 February 1991; revised 23 May 1991. 1991 Mathematics Subject Classification 20F14. Partially funded by an SERC fellowship, whose support is gratefully acknowledged. J. London Math. Soc. (2) 46 (1992) 111-122


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