Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



The Hilbert-Smith conjecture for quasiconformal actions

Author: Gaven J. Martin
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 66-70
MSC (1991): Primary 26A24, 30C60, 53A04, 54F65
Published electronically: May 28, 1999
MathSciNet review: 1694197
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This note announces a proof of the Hilbert-Smith conjecture in the quasiconformal case: A locally compact group $G$ of quasiconformal homeomorphisms acting effectively on a Riemannian manifold is a Lie group. The result established is true in somewhat more generality.

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

  • 1. S. Bochner and D. Montgomery, Locally compact groups of differentiable transformations, Ann. Math. 47 (1946), 639-653. MR 8:253c
  • 2. S. Donaldson and D. Sullivan, Quasiconformal $4$-manifolds, Acta Math. 163 (1989), 181-252. MR 91d:57012
  • 3. D. Hilbert, Mathematische Probleme, Nachr. Akad. Wiss. Göttingen (1900), 253-297.
  • 4. T. Iwaniec and G. J. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), 241-254. MR 97i:30032
  • 5. G. J. Martin, UQR mappings, Siegel's theorem and the Hilbert-Smith conjecture, in preparation.
  • 6. V. Mayer, Uniformly quasiregular mappings of Lattès type, Conformal Geometry and Dynamics 1 (1997), 24-27. MR 98j:30017
  • 7. G.V. Maz'ja, Sobolev spaces, Springer-Verlag, 1985. MR 87g:46056
  • 8. D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York, 1955. MR 17:383b
  • 9. F. Raymond and R. F. Williams, Examples of $p$-adic transformation groups, Ann. Math. 78 (1963), 92-106. MR 27:756
  • 10. D. Repovs and E.V. Scepin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Annalen 2 (1997), 361-364. MR 99c:57066
  • 11. S. Rickman, Quasiregular mappings, Springer-Verlag, 1993. MR 95g:30026
  • 12. D. Sullivan, Quasiconformal and Lipschitz structures, to appear.
  • 13. C-T. Yang, $p$-adic transformation groups, Michigan Math. J. 7 (1960), 201-218. MR 22:11065

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 26A24, 30C60, 53A04, 54F65

Retrieve articles in all journals with MSC (1991): 26A24, 30C60, 53A04, 54F65

Additional Information

Gaven J. Martin
Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand

Received by editor(s): November 9, 1998
Published electronically: May 28, 1999
Additional Notes: Research supported in part by a grant from the N.Z. Marsden Fund.
Communicated by: Walter Neumann
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society