- Title
- Pro-p groups of positive rank gradient and Hausdorff dimension
- Creator
- Ocaña, Oihana Garaialde; Garrido, Alejandra; Klopsch, Benjamin
- Relation
- Journal of the London Mathematical Society Vol. 101, Issue 3, p. 1008-1040
- Publisher Link
- http://dx.doi.org/10.1112/jlms.12295
- Publisher
- Wiley-Blackwell Publishing
- Resource Type
- journal article
- Date
- 2020
- Description
- Using Hausdorff dimension, we show that finitely generated closed subgroups (Formula presented.) of infinite index in a finitely generated pro- (Formula presented.) group (Formula presented.) of positive rank gradient never contain any infinite subgroups (Formula presented.) that are subnormal in (Formula presented.) via finitely generated successive quotients. This generalises similar assertions that were known to hold for non-abelian free pro- (Formula presented.) groups and other related pro- (Formula presented.) groups. Our main results are as follows. We show that every finitely generated pro- (Formula presented.) group (Formula presented.) of positive rank gradient has full Hausdorff spectrum (Formula presented.) with respect to the Frattini series (Formula presented.) (or, more generally, any iterated verbal filtration). Using Lie-theoretic techniques we also prove that finitely generated non-abelian free pro- (Formula presented.) groups and non-soluble Demushkin groups (Formula presented.) have full Hausdorff spectrum (Formula presented.) with respect to the Zassenhaus series (Formula presented.). This resolves a long-standing problem in the subject. In fact, the results hold more generally for finite direct products of finitely generated pro- (Formula presented.) groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro- (Formula presented.) groups and non-soluble Demushkin groups. Finally, we determine the normal Hausdorff spectra of such direct products.
- Subject
- 20E07; 20E18; 20F40 (primary); Hausdorff dimension
- Identifier
- http://hdl.handle.net/1959.13/1447513
- Identifier
- uon:43170
- Identifier
- ISSN:0024-6107
- Language
- eng
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