A pro-p group with infinite normal Hausdorff spectra

Abstract

Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum1] denotes the Hausdorff dimension function associated to the p-power series P : G p i , i ∈ N 0 . More precisely, we show that hspec P (G) = [0, 1 /3] ∪ {1} contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra hspec S (G) with respect to other filtration series S have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower p-series and the modular dimension subgroup series.Lastly, we pin down the ordinary Hausdorff spectra hspec S (G) = {hdim S G (H) | H ≤ c G} with respect to the standard filtration series S. The spectrum hspec L (G) for the lower p-series L displays surprising new features.

A pro‐pgroup with full normal Hausdorff spectra

“…This resolves Problems 1.2 (b),(c) in [7] and Problem 5 in [2] for all five standard series. The latter problem was already solved previously for the series $$\mathcal {D}$$, $$\mathcal {P}$$, $$\mathcal {P}^*$$ and $$\mathcal {F}$$: in [6, VIII, §7] it was seen that $$W \cong C_p\ \hat{\wr }\ \mathbb {Z}_p$$ has $$\operatorname{hspec}^{\mathcal {D}}(W) = \operatorname{hspec}^{\mathcal {P}}(W) = \operatorname{hspec}^{\mathcal {F}}(W) = [0,1]$$, and by completely different means it was shown in [5] that a non‐abelian finitely generated free pro‐ p group E has $$\operatorname{hspec}^{\mathcal {D}}(E) = \operatorname{hspec}^{\mathcal {P}^*}(E) = \operatorname{hspec}^{\mathcal {F}}(E) = [0,1]$$.…”

“…For an infinite countably based pro‐ p group G , equipped with a filtration series $$\mathcal {S} : G = S_0 \supseteq S_1\supseteq \dots$$, and a closed subgroup $$H \le _\mathrm{c} G$$ we adopt the following terminology from [7]: we say that H has strong Hausdorff dimension in G with respect to $$\mathcal {S}$$ if its Hausdorff dimension is given by a proper limit, i.e., if $$\begin{equation*} \operatorname{hdim}^{\mathcal {S}}_G(H) = \lim _{i \rightarrow \infty } \frac{\log _p \vert H S_i : S_i \vert }{\log _p \vert G : S_i \vert }. \end{equation*}$$…”

“…Already twenty years ago, Shalev [10, §4.7] put up the challenge to construct finitely generated pro‐ p groups with infinite normal Hausdorff spectra and he asked whether the normal Hausdorff spectra could even contain infinite real intervals. Recently, Klopsch and Thillaisundaram [7] succeeded in constructing such examples, with respect to the five standard filtration series. Even though the normal Hausdorff spectra of their groups each contain infinite intervals, none of the spectra covers the full unit interval [0,1].…”

Pro‐p groups of positive rank gradient and Hausdorff dimension

“…There is also a natural interest in the finitely generated Hausdorff spectrum of a finitely generated pro‐ group , with respect to a filtration , defined as compare [, § 4.7] and .…”

“…The normal Hausdorff spectrum of a finitely generated pro‐ group , with respect to a filtration , is compare [, § 4.7] and . Theorem Let be a non‐trivial finite direct product of finitely generated pro‐ groups of positive rank gradient.…”