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Abstract
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Using wreath products, we construct a finitely generated
pro-
group
with infinite normal Hausdorff spectrum
here
denotes the Hausdorff dimension function associated to the
-power
series
,
. More precisely,
we show that
contains an infinite interval; this settles a question of Shalev.
Furthermore, we prove that the normal Hausdorff spectra
with respect to other
filtration series
have a similar shape. In particular, our analysis applies to
standard filtration series such as the Frattini series, the lower
-series
and the modular dimension subgroup series.
Lastly, we pin down the ordinary Hausdorff spectra
with respect to the standard filtration
series . The spectrum
for the lower
-series
displays surprising new features.
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Keywords
pro-$p$ groups, Hausdorff dimension, Hausdorff spectrum,
normal Hausdorff spectrum
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Mathematical Subject Classification 2010
Primary: 20E18
Secondary: 28A78
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Milestones
Received: 4 December 2018
Revised: 5 July 2019
Accepted: 6 July 2019
Published: 4 January 2020
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