Benjamin Klopsch: Hausdorff spectra of finitely generated pro-p groups
Every finitely generated pro-p group G comes equipped with a range of translation-invariant metrics that are naturally induced by filtration series such as the p-power filtration or the (modular) dimension subgroup filtration. Given such a metric, the distribution of closed subgroup in G gives rise to a corresponding Hausdorff spectrum. It is a long-standing open question whether the finiteness of the Hausdorff spectrum, with respect to the p-power filtration, say, implies that the pro-p group G is p-adic analytic. After a short general introduction, I will mainly report on joint work with Anitha Thillaisundaram and Amaia Zugadi-Reizabal, highlighting also some open problems arising from our results. Time permitting, I will then talk about ongoing work with Alejandra Garrido and Oihana Garaialde-Ocana concerning the Hausdorff spectra of free pro-p groups.
