Skip to main content
SpringerLink
Log in

The Scale, Tidy Subgroups and Flat Groups

  • Chapter
  • First Online:
2016 MATRIX Annals

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

  • 942 Accesses

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.

Notes prepared by John J. Harrison.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abels, H.: Kompakt definierbare topologische Gruppen. Math. Ann. 197, 221–233 (1972)

    Article  MathSciNet  Google Scholar 

  2. Abramenko, P., Brown, K.S.: Buildings: Theory and Applications. Springer Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)

    Book  Google Scholar 

  3. Baumgartner, U., Willis, G.A.: Contraction groups for automorphisms of totally disconnected groups. Isr. J. Math. 142, 221–248 (2004)

    Article  MathSciNet  Google Scholar 

  4. Baumgartner, U., Willis, G.A.: The direction of an automorphism of a totally disconnected locally compact group. Math. Z. 252, 393–428 (2006)

    Article  MathSciNet  Google Scholar 

  5. Baumgartner, U., Rémy, B., Willis, G.A.: Flat rank of automorphism groups of buildings. Transformation Groups 12, 413–436 (2007)

    Article  MathSciNet  Google Scholar 

  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. Bergman, G.M., Lenstra, Jr., H.W.: Subgroups close to normal subgroups. J. Algebra 127(1), 80–97 (1989)

    Article  MathSciNet  Google Scholar 

  8. Bourbaki, N.: Commutative Algebra. Springer, New York (1989)

    Google Scholar 

  9. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local, I. Données radicielles values. Publ. Math. IHES 41, 5–251 (1972)

    Article  Google Scholar 

  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. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic, New York (1978)

    Google Scholar 

  12. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Springer Science and Business Media, Alemania (1994)

    Book  Google Scholar 

  13. Jaworski, W.: On contraction groups of automorphisms of totally dis-connected locally compact groups. Isr. J. Math. 172, 1–8 (2009)

    Article  MathSciNet  Google Scholar 

  14. Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)

    Book  Google Scholar 

  15. Krön, B., Möller, R.G.: Analogues of Cayley graphs for topological groups. Math. Z. 258, 637–675 (2008)

    Article  MathSciNet  Google Scholar 

  16. Möller, R.G.: Structure theory of totally disconnected locally compact groups via graphs and permutations. Can. J. Math. 54, 795–827 (2002)

    Article  MathSciNet  Google Scholar 

  17. Monod, N.: Continuous bounded cohomology of locally compact groups. Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)

    Google Scholar 

  18. Schlichting, G.: Operationen mit periodischen Stabilisatoren. Arch. Math. 34(2), 97–99 (1980)

    Article  MathSciNet  Google Scholar 

  19. Shalom, Y., Willis, G.A.: Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity. Geom. Funct. Anal. 23, 1631–1683 (2013)

    Article  MathSciNet  Google Scholar 

  20. Stroppel, M.: Locally Compact Groups. European Mathematical Society, Zürich (2006)

    Google Scholar 

  21. Tao, T.: Hilbert’s Fifth Problem and Related Topics. American Mathematical Society, Providence (2014)

    Book  Google Scholar 

  22. Tits, J.: Groupes associés aux algèbres de Kac-Moody, Astérisque, no. 177–178, Exp. No. 700, 7–31; Séminaire Bourbaki, vol. 1988/89

    Google Scholar 

  23. Willis, G.A.: The structure of totally disconnected, locally compact groups. Math. Ann. 300, 341–363 (1994)

    Article  MathSciNet  Google Scholar 

  24. Willis, G.A.: Further properties of the scale function on totally disconnected groups. J. Algebra 237, 142–164 (2001)

    Article  MathSciNet  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  26. Willis, G.A.: The nub of an automorphism of a totally disconnected, locally compact group. Ergodic Theory Dyn. Syst. 34, 1365–1394 (2014)

    Article  MathSciNet  Google Scholar 

  27. Willis, G.A.: The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups. Math. Ann. 361, 403–442 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. University of Newcastle, University Drive, Callaghan, NSW, 2308, Australia

    George Willis

Authors
  1. George Willis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Willis .

Editor information

Editors and Affiliations

  1. University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. University of Western Australia, Crawley, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, USA

    Terence Tao

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Willis, G. (2018). The Scale, Tidy Subgroups and Flat Groups. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_26

Download citation

Publish with us

Policies and ethics

Search

Navigation