Vol. 303, No. 2, 2019

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A pro-$p$ group with infinite normal Hausdorff spectra

Benjamin Klopsch and Anitha Thillaisundaram

Vol. 303 (2019), No. 2, 569–603

Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum

hspecP(G) = {hdim GP(H)H cG};

here hdimGP : {XX G} [0,1] denotes the Hausdorff dimension function associated to the p-power series P : Gpi , i 0. More precisely, we show that hspecP(G) =[0, 1 3] {1} contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra hspecS(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

hspecS(G) = {hdim GS(H)H cG}

with respect to the standard filtration series S. The spectrum hspec(G) for the lower p-series  displays surprising new features.

pro-$p$ groups, Hausdorff dimension, Hausdorff spectrum, normal Hausdorff spectrum
Mathematical Subject Classification 2010
Primary: 20E18
Secondary: 28A78
Received: 4 December 2018
Revised: 5 July 2019
Accepted: 6 July 2019
Published: 4 January 2020
Benjamin Klopsch
Mathematisches Institut
Anitha Thillaisundaram
School of Mathematics and Physics
University of Lincoln
United Kingdom