The Hilbert--Smith conjecture for three-manifolds

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 ${\mathbb{Z}}_p\to \mathrm{MCG}(F)$ gives the desired contradiction. The approach is local on $M$.

24 pages, 1 figure; to appear in Journal of the AMS

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