# A pro-p group with infinite normal Hausdorff spectra

## TL;DR

This paper constructs a finitely generated pro-p group with an infinite normal Hausdorff spectrum, answering a question of Shalev and analyzing its shape across various filtration series.

## Contribution

It introduces a new pro-p group with an infinite normal Hausdorff spectrum and explores its spectral properties across multiple filtration series.

## Key findings

- Normal Hausdorff spectrum contains an infinite interval
- Spectra with respect to various filtration series have similar shapes
- The spectrum for the lower p-series exhibits surprising features

## 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.