# Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group.

Is this conjecture still unsolved? Is there any references to more details?

• I see three references to more details in the article you yourself have linked. – Chris Eagle Mar 17 '13 at 11:01
• These three references are a bit outdated. In february, 2011 Open Problem Garden announced this conjecture as an open problem. I'm looking for More recent attempts to solve this problem. – TXC Mar 17 '13 at 11:17
• Yes it's unsolved. It's known for a while that solving it (say in dimension $\le n$) reduces to proving that for every prime $p$ the group of $p$-adics $\mathbf{Z}_p$ has no continuous faithful action on any topological connected manifold of dimension $n$. I think it's also known to be enough to prove it for connected open subsets of $\mathbf{R}^n$. – YCor Apr 28 '19 at 12:01

According to J. PARDON paper (arXiv:1112.2324) it has been resolved in dimension $3$ but in dimension $\geq 4$ is open still.