The Hilbert–Smith conjecture for three-manifolds

Pardon, John

doi:10.1090/S0894-0347-2013-00766-3

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
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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 $\mathbb{Z}_p$ (the -adic integers) on a connected three-manifold. If $\mathbb{Z}_p$ acts faithfully on , we find an interesting $\mathbb{Z}_p$ -invariant open set $U\subseteq M$ with $H_2(U)=\mathbb{Z}$ 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 $\mathbb{Z}_p$ . An analysis of the resulting homomorphism $\mathbb{Z}_p\to \operatorname {MCG}(F)$ gives the desired contradiction. The approach is local on .

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