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

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 2ᵈ rational numbers. Furthermore, we show that, if G is a countably based pro-p group with an open subgroup mapping onto the free abelian pro-p group Zₚ Zₚ, then for every prescribed finite set \0, 1\ X 0, 1 there is a filtration series S such that the Hausdorff spectrum of G, with respect to S, is X. In particular, the cardinality of the Hausdorff spectrum of G, with respect to S, is unbounded, as S runs through all filtration series of G that result in a finite Hausdorff spectrum. Finally, we establish that finitely generated non-abelian direct products G of free pro-p groups have full Hausdorff spectrum 0, 1 with respect to the lower p-series.