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
DOI:
https://doi.org/10.1090/S1079-6762-99-00062-1
Published electronically:
May 28, 1999
MathSciNet review:
1694197
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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 
of quasiconformal homeomorphisms acting effectively on a Riemannian manifold is a Lie group. The result established is true in somewhat more generality.
- 1. S. Bochner and D. Montgomery, Locally compact groups of differentiable transformations, Ann. Math. 47 (1946), 639-653. MR 8:253c
- 2. S. K. Donaldson and D. P. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163 (1989), no. 3-4, 181–252. MR 1032074, https://doi.org/10.1007/BF02392736
- 3. D. Hilbert, Mathematische Probleme, Nachr. Akad. Wiss. Göttingen (1900), 253-297.
- 4. Tadeusz Iwaniec and Gaven Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 241–254. MR 1404085
- 5. G. J. Martin, UQR mappings, Siegel's theorem and the Hilbert-Smith conjecture, in preparation.
- 6. Volker Mayer, Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), 104–111. MR 1482944, https://doi.org/10.1090/S1088-4173-97-00013-1
- 7. Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985
- 8. D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York, 1955. MR 17:383b
- 9. Frank Raymond and R. F. Williams, Examples of 𝑝-adic transformation groups, Ann. of Math. (2) 78 (1963), 92–106. MR 0150769, https://doi.org/10.2307/1970504
- 10. Dušan Repovš and Evgenij Ščepin, A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps, Math. Ann. 308 (1997), no. 2, 361–364. MR 1464908, https://doi.org/10.1007/s002080050080
- 11. Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941
- 12. D. Sullivan, Quasiconformal and Lipschitz structures, to appear.
- 13. Chung-Tao Yang, 𝑝-adic transformation groups, Michigan Math. J. 7 (1960), 201–218. MR 0120310
-manifolds, Acta Math. 163 (1989), 181-252. 
-adic transformation groups, Ann. Math. 78 (1963), 92-106. 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
Email:
martin@math.auckland.ac.nz
DOI:
https://doi.org/10.1090/S1079-6762-99-00062-1
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