Using Hausdorff dimension, we show that finitely generated closed subgroups H of infinite index in a finitely generated pro-p group G of positive rank gradient never contain any infinite subgroups K that are subnormal in G via finitely generated successive quotients. This generalises similar assertions that were known to hold for non-abelian free pro-p groups and other related pro-p groups.Our main results are as follows. We show that every finitely generated pro-p group G of positive rank gradient has full Hausdorff spectrum hspec F (G) = [0, 1] with respect to the Frattini series F (or, more generally, any iterated verbal filtration). Using Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-p groups and non-soluble Demushkin groups G have full Hausdorff spectrum hspec Z (G) = [0, 1] with respect to the Zassenhaus series Z. This resolves a long-standing problem in the subject.In fact, the results hold more generally for finite direct products of finitely generated prop groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-p groups and non-soluble Demushkin groups.Finally, we determine the normal Hausdorff spectra of such direct products.
A decade ago, J. F. Carlson proved that there are finitely many cohomology rings of finite
2
2
-groups of fixed coclass, and he conjectured that this result ought to be true for odd primes. In this paper, we prove the non-twisted case of Carlson’s conjecture for any prime and we show how to proceed in the twisted case.
Abstract. Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-p groups of nilpotency class smaller than p and finitely generated nilpotent Zp-Lie algebras of nilpotency class smaller than p. Denote by H iGr and H i Lie the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for i = 0, 1 and 2, and for a given category of modules the cohomology functors H i Gr • exp and H i Lie are naturally equivalent. A similar result is proven for i = 3 and the relative cohomology groups.
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