Elementary 𝑝-adic Lie groups have finite construction rank

Glöckner, Helge

doi:10.1090/proc/13637

Elementary $p$-adic Lie groups have finite construction rank
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by Helge Glöckner

Proc. Amer. Math. Soc. 145 (2017), 5007-5021

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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