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Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.

Is it true that the tensor product $\mathbb Z_p[[G]]\otimes_{\mathbb Z_p[\Gamma]} \mathbb Z_p[[G]]$ is torsion-free as an abelian group?

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  • $\begingroup$ The completed group algebra is by definition the projective limit of $\mathbf{Z}_p[G/U]$ where $U$ ranges over [open] finite index subgroups, is that correct? $\endgroup$
    – YCor
    May 17, 2018 at 23:23
  • $\begingroup$ @Ycor yes, that's true. $\endgroup$
    – Pablo
    May 18, 2018 at 9:37
  • $\begingroup$ A trivial remark is that the question is equivalent to asking whether it has a no element of order $p$ (since this is a $\mathbf{Z}_p$-module). $\endgroup$
    – YCor
    May 18, 2018 at 9:44

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