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 spectrum with respect to the p-power series. More precisely, we show that this normal Hausdorff spectrum contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra of G 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 p-series and the modular dimension subgroup series. Lastly, we pin down the ordinary Hausdorff spectra of G with respect to the standard filtration series. The spectrum of G for the lower p-series displays surprising new features.