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


Author: Helge Glöckner
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
Published electronically: July 10, 2017
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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
Email: glockner@math.upb.de

DOI: https://doi.org/10.1090/proc/13637
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
Article copyright: © Copyright 2017 American Mathematical Society