On the Lie theory of p-adic analytic groups

The aim of this paper is to fill a small, but fundamental, gap in the theory of p-adic analytic groups. We illustrate by example that the now standard notion of a uniformly powerful pro-p group is more restrictive than Lazard’s concept of a saturable pro-p group. For instance, the Sylow-pro-p subgroups of many classical groups are saturable, but need not be uniformly powerful. Extending work of Ilani, we obtain a correspondence between subgroups and Lie sublattices of saturable pro-p groups. This leads to applications, for instance, in the subject of subgroup growth.