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Journal of the London Mathematical Society 1992 s2-46(1):111-122; doi:10.1112/jlms/s2-46.1.111
© 1992 by London Mathematical Society
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© Oxford University Press

Growth Functions, p-Adic Analytic Groups, and Groups of Finite Coclass

Aner Shalev

School of Mathematical Sciences, Queen Mary and Westfield College University of London, Mile End Road, London E1 4NS

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 < 1/8 and for all sufficiently large k, G has at most pckk open subgroups of index pk, then G is p-adic analytic; (2) if, for some k, (G:Gpk < ppk+k–1 then G is p-acid analytic; (3) if, for some n, (G:Gn) < Formula 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.

Current address: Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel


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