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Recall that an infinite pro-$p$ group $G$ is called just-infinite if all non-trivial closed normal subgroup of $G$ have finite index.

Question: Let $G$ be an insoluble $p$-adic analytic just-infinite pro-$p$ group. Is it true that $G$ is torsion-free?

If $\text{dim}~G<p-1$, then it is true, cf. Theorem 1.3 in [Klopsch, Benjamin. "On the Lie theory of p-adic analytic groups." Mathematische Zeitschrift 249 (2005): 713-730.] Any references or counterexample will be appreciated.

Added. Let $p$ be an odd prime. The following are equivalent:

  • (1) Any insoluble $p$-adic analytic just-infinite pro-$p$ group is torsion-free.

  • (2) Any Sylow-pro-$p$ subgroup of ${\rm SL}_n(\mathbf{Z}_p)$ is torsion-free for any $n\geq 2$.

Sketch of proof:

  • $(1)\implies (2)$: Note that any Sylow-pro-$p$ subgroup of ${\rm SL}_n(\mathbf{Z}_p)$ is an insoluble $p$-adic analytic just-infinite pro-$p$ group;

  • $(2)\implies (1)$: Let $G$ be any insoluble $p$-adic analytic just-infinite pro-$p$ group. Then $G$ is isomorphic to a closed subgroup of ${\rm GL}_n(\mathbf{Z}_p)$ for some positive integer $n$. Since the abelianization of $G$ is a finite $p$-group, $G$ is isomorphic to a closed subgroup of ${\rm SL}_n(\mathbf{Z}_p)$. Thus, $G$ must be torsion-free.

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  • $\begingroup$ Do you know the answer in the various cases where $G$ is chosen as an open subgroup of $\mathrm{SL}_n(\mathbf{Z}_p)$ for $n\ge 2$? $\endgroup$
    – YCor
    Feb 25, 2023 at 8:20
  • $\begingroup$ When $n\geq 2$ and $p>n+1$, then any open pro-$p$ subgroup of ${\rm SL}_n(\mathbf{Z}_p)$ is torsion-free. $\endgroup$
    – stupid boy
    Feb 25, 2023 at 15:48
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    $\begingroup$ Have you thought about the case $p=n$? $\endgroup$ Mar 1, 2023 at 16:26
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    $\begingroup$ Think about permutation matrices. $\endgroup$ Mar 1, 2023 at 16:49
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    $\begingroup$ Your example generalises for any p=n. $\endgroup$ Mar 1, 2023 at 20:35

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