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 rk(G) can be associated, its construction rank. By a structure theorem of Phillip Wesolek, elementary p-padic 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. Results concerning general p-adic Lie groups are also obtained, concerning the isomorphism types of subquotients needed to build up the latter.