Skip to main content

Endomorphisms of Lie Groups over Local Fields

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

Abstract

Lie groups over totally disconnected local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-72299-3_6
  • Chapter length: 65 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-72299-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Hardcover Book
USD   169.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baumgartner, U., Willis, G.A.: Contraction groups and scales of automorphisms of totally disconnected locally compact groups. Isr. J. Math. 142, 221–248 (2004)

    MathSciNet  CrossRef  Google Scholar 

  2. Bertram, W., Glöckner, H., Neeb, K.-H.: Differential calculus over general base fields and rings. Expo. Math. 22, 213–282 (2004)

    MathSciNet  CrossRef  Google Scholar 

  3. Bhattacharjee, M., MacPherson, D.: Strange permutation representations of free groups. J. Aust. Math. Soc. 74, 267–285 (2003)

    MathSciNet  CrossRef  Google Scholar 

  4. Borel, A.: Linear Algebraic Groups. Springer, New York (1991)

    Google Scholar 

  5. Bourbaki, N.: Variétés différentielles et analytiques. Fascicule de résultats. Hermann, Paris (1967)

    Google Scholar 

  6. Bourbaki, N.: Topological Vector Spaces, chaps. 1–5. Springer, Berlin (1987)

    CrossRef  Google Scholar 

  7. Bourbaki, N.: Lie Groups and Lie Algebras, chaps. 1–3. Springer, Berlin (1989)

    Google Scholar 

  8. Bruhat, F., Tits, J.: Groupes reductifs sur un corps local. Publ. Math. Inst. Hautes Étud. Sci. 41, 5–251 (1972)

    CrossRef  Google Scholar 

  9. Bywaters, T.P., Glöckner, H., Tornier, S., Contraction groups and passage to subgroups and quotients for endomorphisms of totally disconnected locally compact groups. Israel J. Math. (cf. arXiv:1612.06958) (to appear)

    Google Scholar 

  10. Dani, S.G., Shah, R.: Contraction subgroups and semistable measures on p-adic Lie groups. Math. Proc. Camb. Philos. Soc. 110, 299–306 (1991)

    MathSciNet  CrossRef  Google Scholar 

  11. Dani, S.G., Shah, N.A., Willis, G.A.: Locally compact groups with dense orbits under \({\mathbb Z}^d\)-actions by automorphisms. Ergodic Theory Dyn. Syst. 26, 1443–1465 (2006)

    Google Scholar 

  12. Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic Pro-p Groups. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  13. du Sautoy, M., Mann, A., Segal, D.: New Horizons in Pro-p Groups. Birkhäuser, Boston (2000)

    Google Scholar 

  14. Giordano Bruno, A., Virili, S.: Topological entropy in totally disconnected locally compact groups. Ergod. Theory Dyn. Syst. 37, 2163–2186 (2017)

    MathSciNet  CrossRef  Google Scholar 

  15. Glöckner, H.: Scale functions on p-adic Lie groups. Manuscripta Math. 97, 205–215 (1998)

    Google Scholar 

  16. Glöckner, H.: Implicit functions from topological vector spaces to Banach spaces. Isr. J. Math. 155, 205–252 (2006)

    MathSciNet  CrossRef  Google Scholar 

  17. Glöckner, H.: Contractible Lie groups over local fields. Math. Z. 260, 889–904 (2008)

    MathSciNet  CrossRef  Google Scholar 

  18. Glöckner, H.: Invariant manifolds for analytic dynamical systems over ultrametric fields. Expo. Math. 31, 116–150 (2013)

    MathSciNet  CrossRef  Google Scholar 

  19. Glöckner, H.: Invariant manifolds for finite-dimensional non-archimedean dynamical systems. In: Glöckner, H., Escassut, A., Shamseddine, K. (eds.) Advances in Non-Archimedean Analysis, pp. 73–90. Contemporary Mathematics, vol. 665. American Mathematical Society, Providence, RI (2016)

    Google Scholar 

  20. Glöckner, H.: Lie groups over non-discrete topological fields. Preprint (2004). arXiv:math/0408008

    Google Scholar 

  21. Glöckner, H.: Lectures on Lie groups over local fields. In: Caprace, P.-E., Monod, N. (eds.) New Directions in Locally Compact Groups. London Math. Soc. Lect. Notes Series 447 (to appear)

    Google Scholar 

  22. Glöckner, H.: Contraction groups of analytic endomorphisms and dynamics on the big cell. University of Paderborn (2017)

    Google Scholar 

  23. Glöckner, H., Raja, C.R.E.: Expansive automorphisms of totally disconnected, locally compact groups. J. Group Theory 20, 589–619 (2017)

    Google Scholar 

  24. Glöckner, H., Willis, G.A.: Uniscalar p-adic Lie groups. Forum Math. 13, 413–421 (2001)

    Google Scholar 

  25. Glöckner, H., Willis, G.A.: Directions of automorphisms of Lie groups over local fields compared to the directions of Lie algebra automorphisms. Topol. Proc. 31, 481–501 (2007)

    Google Scholar 

  26. Glöckner, H., Willis, G.A.: Classification of the simple factors appearing in composition series of totally disconnected contraction groups. J. Reine Angew. Math. 634, 141–169 (2010)

    Google Scholar 

  27. Hazod, W., Siebert, E.: Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups. Kluwer, Dordrecht (2001)

    CrossRef  Google Scholar 

  28. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis I. Springer, Berlin (1963)

    MATH  Google Scholar 

  29. Hofmann, K.H., Morris, S.A.: The Structure of Compact Groups. de Gruyter, Berlin (1998)

    Google Scholar 

  30. Humphreys, J.E.: Linear Algebraic Groups. Springer, New York (1975)

    Google Scholar 

  31. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, Berlin (1994)

    Google Scholar 

  32. Irwin, M.C.: On the stable manifold theorem. Bull. Lond. Math. Soc. 2, 196–198 (1970)

    MathSciNet  CrossRef  Google Scholar 

  33. Jacobson, N.: A note on automorphisms and derivations of Lie algebras. Proc. Am. Math. Soc. 6, 281–283 (1955)

    MathSciNet  CrossRef  Google Scholar 

  34. Jacobson, N.: Basic Algebra II. W. H. Freeman and Company, New York (1989)

    Google Scholar 

  35. Jaikin-Zapirain, A., Klopsch, B.: Analytic groups over general pro-p domains. J. Lond. Math. Soc. 76, 365–383 (2007)

    Google Scholar 

  36. Jaworski, W.: On contraction groups of automorphisms of totally disconnected locally compact groups. Isr. J. Math. 172, 1–8 (2009)

    MathSciNet  CrossRef  Google Scholar 

  37. Kepert, A., Willis, G.A.: Scale functions and tree ends. J. Aust. Math. Soc. 70, 273–292 (2001)

    MathSciNet  CrossRef  Google Scholar 

  38. Lazard, M.: Groupes analytiques p-adiques. IHES Publ. Math. 26, 389–603 (1965)

    Google Scholar 

  39. Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Springer, Berlin (1991)

    CrossRef  Google Scholar 

  40. Palmer, T. W.: Banach Algebras and the General Theory of ∗-Algebras, vol. 2. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  41. Parreau, A.: Sous-groupes elliptiques de groupes linéaires sur un corps valué. J. Lie Theory 13, 271–278 (2003)

    Google Scholar 

  42. Raja, C.R.E.: On classes of p-adic Lie groups. New York J. Math. 5, 101–105 (1999)

    Google Scholar 

  43. Raja, C.R.E., Shah, R.: Some properties of distal actions on locally compact groups. Ergodic Theory Dyn. Syst., doi: 10.1017/etds.2017.58 (to appear)

    Google Scholar 

  44. Rathai, N.: Endomorphismen total unzusammenhängender lokal kompakter Gruppen. Bachelor’s thesis, Universität Paderborn (2016)

    Google Scholar 

  45. Reid, C.D.: Endomorphisms of profinite groups. Groups Geom. Dyn. 8, 553–564 (2014)

    MathSciNet  CrossRef  Google Scholar 

  46. Reid, C.D.: Distal actions on coset spaces in totally disconnected, locally compact groups. Preprint (2016). arXiv:1610.06696

    Google Scholar 

  47. Schikhof, W.H.: Ultrametric Calculus. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  48. Schneider, P.: p-Adic Lie Groups. Springer, Berlin (2011)

    Google Scholar 

  49. Serre, J.-P.: Lie Algebras and Lie Groups. Springer, Berlin (1992)

    CrossRef  Google Scholar 

  50. Siebert, E.: Contractive automorphisms of locally compact groups. Math. Z. 191, 73–90 (1986)

    MathSciNet  CrossRef  Google Scholar 

  51. Siebert, E.: Semistable convolution semigroups and the topology of contraction groups. In: Heyer, H. (ed.) Probability Measures on Groups IX. Lecture Notes in Mathematics, vol. 1379, pp. 325–343. Springer, Berlin (1989)

    CrossRef  Google Scholar 

  52. Springer, T.A.: Linear Algebraic Groups. Birkhäuser, Boston (1998)

    CrossRef  Google Scholar 

  53. Stroppel, M.: Locally Compact Groups. EMS Publishing House, Zurich (2006)

    CrossRef  Google Scholar 

  54. Wang, J.S.P.: The Mautner phenomenon for p-adic Lie groups. Math. Z. 185, 403–412 (1984)

    MathSciNet  CrossRef  Google Scholar 

  55. Weil, A.: Basic Number Theory. Springer, New York (1967)

    CrossRef  Google Scholar 

  56. Wells, J.C.: Invariant manifolds of non-linear operators. Pac. J. Math. 62, 285–293 (1976)

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

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

    MathSciNet  CrossRef  Google Scholar 

  61. Wilson, J.S.: Profinite Groups. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful for the support provided by the University of Melbourne (Matrix Center, Creswick) and the University of Newcastle (NSW), notably George A. Willis, which enabled participation in the ‘Winter of Disconnectedness.’ A former unpublished manuscript concerning the scale of automorphisms dating back to 2006 was supported by DFG grant 447 AUS-113/22/0-1 and ARC grant LX 0349209.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Helge Glöckner .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Glöckner, H. (2018). Endomorphisms of Lie Groups over Local Fields. 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_6

Download citation