Pro-$p$ groups of positive rank gradient and Hausdorff dimension

Abstract: Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$ that are subnormal in~$G$ via finitely generated successive quotients. This pro-$p$ version of a well-known theorem of Greenberg generalises similar assertions that were known to hold for non-abelian free pro-$p$ groups, non-soluble Demushkin pro-$p$ groups and other related pro-$p$ groups. The result we prove is reminiscent of Gaboriau's theorem for countable groups with positive first $\ell^2$-Betti number, but not quite a direct analogue. The approach via the notion of Hausdorff dimension in pro-$p$ groups also leads to our main results. We show that every finitely generated pro-$p$ group $G$ of positive rank gradient has full Hausdorff spectrum $\text{hspec}^\mathcal{F}(G) = [0,1]$ with respect to the Frattini series~$\mathcal{F}$. Using different, 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 $\text{hspec}^\mathcal{Z}(G) = [0,1]$ with respect to the Zassenhaus series~$\mathcal{Z}$. This resolves a long-standing problem in the subject of Hausdorff dimensions in pro-$p$ groups. The results about full Hausdorff spectra hold more generally for finite direct products of finitely generated pro-$p$ 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, respectively. Analogously, the aforementioned results with respect to the Frattini series generalise further to Hausdorff dimension functions with respect to arbitrary iterated verbal filtrations. Finally, we determine the normal Hausdorff spectra of such direct products.