The $p$-adic formulation of the Hilbert–Smith conjecture

Let ${\mathbb{Z}}_p$ denote the topological group of $p$-adic integers, and let $X$ be a connected $m$-manifold. Hilbert–Smith conjecture, $p$-adic formulation. The group ${\mathbb{Z}}_p$ does not act faithfully on $X$. This formulation is stated in the paper as equivalent to the locally compact-group formulation. It is known in dimensions $1$, $2$, and $3$, while the general case remains open.