## The Hilbert–Smith conjecture for three-manifolds

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- by John Pardon PDF
- J. Amer. Math. Soc.
**26**(2013), 879-899 Request permission

## 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 $p$-adic integers) on a connected three-manifold. If $\mathbb Z_p$ acts faithfully on $M^3$, 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 $U$ representing a generator of $H_2(U)$. It turns out that there must be one such incompressible surface, say $F$, 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 $M$.## References

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## Additional Information

**John Pardon**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 857067
- Email: pardon@math.stanford.edu
- 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.
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3037790