A pro-2 group with full normal Hausdorff spectra

TL;DR

This paper constructs a specific pro-2 group with a full spectrum of Hausdorff dimensions across multiple filtration series, answering an open question for the prime 2 case and providing new examples in the field.

Contribution

It introduces the first known finitely generated pro-2 group with full Hausdorff spectrum for the lower 2-series, extending previous results to the prime 2 case.

Findings

Constructed a 2-generated pro-2 group with full spectrum [0,1]

Achieved full Hausdorff spectrum with respect to four standard series

Provided the first example for the lower 2-series spectrum in finitely generated pro-2 groups

Abstract

We construct a $2$-generated pro-$2$ group with full normal Hausdorff spectrum $[0,1]$, with respect to each of the four standard filtration series: the $2$-power series, the lower $2$-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-$2$ group with full Hausdorff spectrum with respect to the lower $2$-series.