Title: Hausdorff dimensions in $p$-adic analytic groups,
Abstract: Let $G$ be a finitely generated pro-$p$ group, equipped with the $p$-power
series. The associated metric and Hausdorff dimension function give rise to the
Hausdorff spectrum, which consists of the Hausdorff dimensions of closed
subgroups of $G$. In the case where $G$ is $p$-adic analytic, the Hausdorff
dimension function is well understood; in particular, the Hausdorff spectrum
consists of finitely many rational numbers closely linked to the analytic
dimensions of subgroups of $G$.
Conversely, it is a long-standing open question whether the finiteness of the
Hausdorff spectrum implies that $G$ is $p$-adic analytic. We prove that the
answer is yes, in a strong sense, under the extra condition that $G$ is
soluble.
Furthermore, we explore the problem and related questions also for other
filtration series, such as the lower $p$-series, the Frattini series, the
modular dimension subgroup series and quite general filtration series. For
instance, we prove, for odd primes $p$, that every countably based pro-$p$
group $G$ with an open subgroup mapping onto 2 copies of the $p$-adic integers
admits a filtration series such that the corresponding Hausdorff spectrum
contains an infinite real interval.