Elementary p-adic Lie groups have finite construction rank

TL;DR

This paper proves that elementary p-adic Lie groups have finite construction rank and characterizes their subquotients, contributing to the understanding of their structure within p-adic Lie groups.

Contribution

It establishes that all elementary p-adic Lie groups possess finite construction rank and provides a characterization of their subquotients.

Findings

Elementary p-adic Lie groups have finite construction rank.

Characterization of subquotients for elementary p-adic Lie groups.

Results on subquotients of general p-adic Lie groups.

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.