  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 $M$.
