|
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  of quasiconformal homeomorphisms acting effectively on a Riemannian manifold is a Lie group. The result established is true in somewhat more generality.
- 1.
Salomon
Bochner and Deane
Montgomery, Locally compact groups of differentiable
transformations, Ann. of Math. (2) 47 (1946),
639–653. MR 0018187
(8,253c)
- 2.
S.
K. Donaldson and D.
P. Sullivan, Quasiconformal 4-manifolds, Acta Math.
163 (1989), no. 3-4, 181–252. MR 1032074
(91d:57012), http://dx.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
(97i:30032)
- 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
(electronic). MR
1482944 (98j:30017), http://dx.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
(87g:46056)
- 8.
Deane
Montgomery and Leo
Zippin, Topological transformation groups, Interscience
Publishers, New York-London, 1955. MR 0073104
(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
(27 #756)
- 10.
Dusan
Repovs and Evgenij
Scepin, A proof of the Hilbert-Smith conjecture for actions by
Lipschitz maps, Math. Ann. 308 (1997), no. 2,
361–364. MR 1464908
(99c:57066), http://dx.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
(95g:30026)
- 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
(22 #11065)
- 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
 -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
 -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, -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
Email:
martin@math.auckland.ac.nz
DOI:
http://dx.doi.org/10.1090/S1079-6762-99-00062-1
PII:
S 1079-6762(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
|
