The lower $p$-series of analytic pro-$p$ groups and Hausdorff dimension

Abstract

Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set of rational invariants, canonically associated to $G$, that yield the aforementioned uniform bound and that restrict the possible values for the Hausdorff dimensions of closed subgroups of $G$ with respect to the lower $p$-series. In particular, the Hausdorff spectrum of $G$ with respect to the lower $p$-series is discrete and consists of at most $2^d$ rational numbers.

23 pages; this is the Author Accepted Manuscript version of the following article: I. de las Heras, B. Klopsch and A. Thillaisundaram, The lower p-series of analytic pro-p groups and Hausdorff dimension, J. Lond. Math. Soc. 112 (2) (2025), which has been published in final form at https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.70271