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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Elementary $p$-adic Lie groups have finite construction rank
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by Helge Glöckner PDF
Proc. Amer. Math. Soc. 145 (2017), 5007-5021 Request permission


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:
  • 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:
  • MathSciNet review: 3692013