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

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## 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.

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## Additional Information

**Helge Glöckner**- Affiliation: Universität Paderborn, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany
- MR Author ID: 614241
- Email: glockner@math.upb.de
- Received by editor(s): February 19, 2014
- Received by editor(s) in revised form: February 21, 2014, December 22, 2014, October 21, 2016, and December 20, 2016
- Published electronically: July 10, 2017
- Communicated by: Lev Borisov
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 5007-5021 - MSC (2010): Primary 22E20; Secondary 22E35, 22E46, 22E50
- DOI: https://doi.org/10.1090/proc/13637
- MathSciNet review: 3692013