The Scale, Tidy Subgroups and Flat Groups

Chapter
Part of the MATRIX Book Series book series (MXBS, volume 1)

Abstract

These notes discuss the scale, tidy subgroups, subgroups associated with endomorphisms and flat groups on totally disconnected locally compact (t.d.l.c) groups. The first section discusses the structure theory of subgroups which are minimizing for an endomorphism and introduces the scale of an endomorphism. The second section discusses the applications and properties of the scale function. Section 3 discusses other subgroups which may be associated with endomorphisms in a unique way. Section 4 discusses flat groups of automorphisms, the flat rank and various results about flat groups. The final section discusses the geometry of t.d.l.c groups.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We would like to thank Stephan Tornier, who prepared some of the figures used in these lecture notes.

References

  1. 1.
    Abels, H.: Kompakt definierbare topologische Gruppen. Math. Ann. 197, 221–233 (1972)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abramenko, P., Brown, K.S.: Buildings: Theory and Applications. Springer Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)CrossRefGoogle Scholar
  3. 3.
    Baumgartner, U., Willis, G.A.: Contraction groups for automorphisms of totally disconnected groups. Isr. J. Math. 142, 221–248 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baumgartner, U., Willis, G.A.: The direction of an automorphism of a totally disconnected locally compact group. Math. Z. 252, 393–428 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baumgartner, U., Rémy, B., Willis, G.A.: Flat rank of automorphism groups of buildings. Transformation Groups 12, 413–436 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baumgartner, U., Schlichting, G., Willis, G.A.: Geometric characterization of flat groups of automorphisms. Groups Geom. Dyn. 4, 1–13 (2010)Google Scholar
  7. 7.
    Bergman, G.M., Lenstra, Jr., H.W.: Subgroups close to normal subgroups. J. Algebra 127(1), 80–97 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bourbaki, N.: Commutative Algebra. Springer, New York (1989)Google Scholar
  9. 9.
    Bruhat, F., Tits, J.: Groupes réductifs sur un corps local, I. Données radicielles values. Publ. Math. IHES 41, 5–251 (1972)CrossRefGoogle Scholar
  10. 10.
    Glöckner, H., Willis, G.A.: Classification of the simple factors appearing in composition series of totally disconnected contraction groups. J. Reine Angew. Math. 643, 141–169 (2010)Google Scholar
  11. 11.
    Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic, New York (1978)Google Scholar
  12. 12.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Springer Science and Business Media, Alemania (1994)CrossRefGoogle Scholar
  13. 13.
    Jaworski, W.: On contraction groups of automorphisms of totally dis-connected locally compact groups. Isr. J. Math. 172, 1–8 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)CrossRefGoogle Scholar
  15. 15.
    Krön, B., Möller, R.G.: Analogues of Cayley graphs for topological groups. Math. Z. 258, 637–675 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Möller, R.G.: Structure theory of totally disconnected locally compact groups via graphs and permutations. Can. J. Math. 54, 795–827 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Monod, N.: Continuous bounded cohomology of locally compact groups. Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)Google Scholar
  18. 18.
    Schlichting, G.: Operationen mit periodischen Stabilisatoren. Arch. Math. 34(2), 97–99 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shalom, Y., Willis, G.A.: Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity. Geom. Funct. Anal. 23, 1631–1683 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Stroppel, M.: Locally Compact Groups. European Mathematical Society, Zürich (2006)Google Scholar
  21. 21.
    Tao, T.: Hilbert’s Fifth Problem and Related Topics. American Mathematical Society, Providence (2014)CrossRefGoogle Scholar
  22. 22.
    Tits, J.: Groupes associés aux algèbres de Kac-Moody, Astérisque, no. 177–178, Exp. No. 700, 7–31; Séminaire Bourbaki, vol. 1988/89Google Scholar
  23. 23.
    Willis, G.A.: The structure of totally disconnected, locally compact groups. Math. Ann. 300, 341–363 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Willis, G.A.: Further properties of the scale function on totally disconnected groups. J. Algebra 237, 142–164 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Willis, G.A.: Tidy subgroups for commuting automorphisms of totally disconnected groups: an analogue of simultaneous triangularisation of matrices. N. Y. J. Math. 10, 1–35 (2004). Available at http://nyjm.albany.edu:8000/j/2004/Vol10.htm MathSciNetMATHGoogle Scholar
  26. 26.
    Willis, G.A.: The nub of an automorphism of a totally disconnected, locally compact group. Ergodic Theory Dyn. Syst. 34, 1365–1394 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Willis, G.A.: The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups. Math. Ann. 361, 403–442 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of NewcastleCallaghanAustralia

Personalised recommendations