Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Hyman Bass, Groups of integral representation type, Pacific J. Math. 86 (1980), no. 1, 15-51. MR 586867
  • [2] Udo Baumgartner and George A. Willis, Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221-248. MR 2085717, https://doi.org/10.1007/BF02771534
  • [3] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1-3, translated from the French, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Reprint of the 1975 edition. MR 979493
  • [4] Pierre-Emmanuel Caprace, Colin D. Reid, and George A. Willis, Limits of contraction groups and the Tits core, J. Lie Theory 24 (2014), no. 4, 957-967. MR 3328731
  • [5] Raf Cluckers, Yves Cornulier, Nicolas Louvet, Romain Tessera, and Alain Valette, The Howe-Moore property for real and $ p$-adic groups, Math. Scand. 109 (2011), no. 2, 201-224. MR 2854688, https://doi.org/10.7146/math.scand.a-15185
  • [6] Helge Glöckner, Scale functions on $ p$-adic Lie groups, Manuscripta Math. 97 (1998), no. 2, 205-215. MR 1651404, https://doi.org/10.1007/s002290050097
  • [7] Helge Glöckner, The kernel of the adjoint representation of a $ p$-adic Lie group need not have an abelian open normal subgroup, Comm. Algebra 44 (2016), no. 7, 2981-2988. MR 3507164, https://doi.org/10.1080/00927872.2015.1065859
  • [8] James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
  • [9] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825
  • [10] Jean-Pierre Serre, Lie algebras and Lie groups, 1964 lectures given at Harvard University, 2nd ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992. MR 1176100
  • [11] John S. P. Wang, The Mautner phenomenon for $ p$-adic Lie groups, Math. Z. 185 (1984), no. 3, 403-412. MR 731685, https://doi.org/10.1007/BF01215048
  • [12] Phillip Wesolek, Elementary totally disconnected locally compact groups, Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1387-1434. MR 3356810, https://doi.org/10.1112/plms/pdv013
  • [13] Phillip Wesolek, Totally disconnected locally compact groups locally of finite rank, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 3, 505-530. MR 3335425, https://doi.org/10.1017/S0305004115000122
  • [14] G. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), no. 2, 341-363. MR 1299067, https://doi.org/10.1007/BF01450491

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 22E20, 22E35, 22E46, 22E50

Retrieve articles in all journals with MSC (2010): 22E20, 22E35, 22E46, 22E50


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

American Mathematical Society