The Hilbert-Smith conjecture for three-manifolds
Author:
John Pardon
Journal:
J. Amer. Math. Soc. 26 (2013), 879-899
MSC (2010):
Primary 57S10, 57M60, 20F34, 57S05, 57N10; Secondary 54H15, 55M35, 57S17
DOI:
https://doi.org/10.1090/S0894-0347-2013-00766-3
Published electronically:
March 19, 2013
MathSciNet review:
3037790
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of (the
-adic integers) on a connected three-manifold. If
acts faithfully on
, we find an interesting
-invariant open set
with
and analyze the incompressible surfaces in
representing a generator of
. It turns out that there must be one such incompressible surface, say
, whose isotopy class is fixed by
. An analysis of the resulting homomorphism
gives the desired contradiction. The approach is local on
.
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Additional Information
John Pardon
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305
Email:
pardon@math.stanford.edu
DOI:
https://doi.org/10.1090/S0894-0347-2013-00766-3
Received by editor(s):
April 10, 2012
Received by editor(s) in revised form:
October 27, 2012, and November 25, 2012
Published electronically:
March 19, 2013
Additional Notes:
The author was partially supported by a National Science Foundation Graduate Research Fellowship under grant number DGE–1147470.
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.