On subgroups of demushkin groups

Andozhskii, I. V.

doi:10.1007/bf01116451

Cited by 1 publication

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“…Now, if

G

is free, then so is each

G_{i}

; hence

G_{i}

maps homomorphically onto the direct product of

n

copies of

H \cap G_{i}

. It is known that, if

G

is a non‐soluble Demushkin group, then so is

G_{i}

; compare . In this case,

G_{i}

maps homomorphically onto a free pro‐

p

group of rank

⌊ d false(G_{i} false) / 2 ⌋

and consequently onto the direct product of

⌊ n / 2 ⌋ - 1

copies of

H \cap G_{i}

; compare [, Theorem 12.3.1] and .…”

Section: Hausdorff Spectrum With Respect To Iterated Filtration Seriesmentioning

confidence: 99%

“…First suppose that H ≤ c G is finitely generated and that |G : H| = ∞. Lemma 2.1 ensures that for each n ∈ N, there exists i [14,3,49]. In this case G i maps homomorphically onto a free pro-p group of rank ⌊d(G i )/2⌋ and consequently onto the direct product of ⌊n/2⌋ − 1 copies of H ∩G i ; compare [55, Thm.…”

Section: Hausdorff Spectrum With Respect To Iterated Filtration Seriesmentioning

confidence: 99%

See 1 more Smart Citation

Pro‐p groups of positive rank gradient and Hausdorff dimension

Ocaña

Garrido

Klopsch

2019

J. London Math. Soc.

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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.

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“…Now, if