Elementary $p$-adic Lie groups have finite construction rank

Abstract:

The class of elementary totally disconnected groups is the smallest class of totally disconnected, locally compact, second countable groups which contains all discrete countable groups, all metrizable pro-finite groups, and is closed under extensions and countable ascending unions. To each elementary group $G$, a (possibly infinite) ordinal number $\operatorname {rk}(G)$ can be associated, its construction rank. By a structure theorem of Phillip Wesolek, elementary $p$-adic Lie groups are among the basic building blocks for general $\sigma$-compact $p$-adic Lie groups. We characterize elementary $p$-adic Lie groups in terms of the subquotients needed to describe them. The characterization implies that every elementary $p$-adic Lie group has finite construction rank. Structure theorems concerning general $p$-adic Lie groups are also obtained.