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: 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.
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- S. Donaldson and D. Sullivan, Quasiconformal $4$-manifolds, Acta Math. 163 (1989), 181–252.
- D. Hilbert, Mathematische Probleme, Nachr. Akad. Wiss. Göttingen (1900), 253–297.
- T. Iwaniec and G. J. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), 241–254.
- G. J. Martin, UQR mappings, Siegel’s theorem and the Hilbert–Smith conjecture, in preparation.
- V. Mayer, Uniformly quasiregular mappings of Lattès type, Conformal Geometry and Dynamics 1 (1997), 24–27.
- G.V. Maz’ja, Sobolev spaces, Springer-Verlag, 1985.
- D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York, 1955.
- F. Raymond and R. F. Williams, Examples of $p$-adic transformation groups, Ann. Math. 78 (1963), 92–106.
- D. Repovš and E.V. Ščepin, A proof of the Hilbert–Smith conjecture for actions by Lipschitz maps, Math. Annalen 2 (1997), 361–364.
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Additional Information
Gaven J. Martin
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
MR Author ID:
120465
Email:
martin@math.auckland.ac.nz
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
