Torsion in a tensor product over a group ring

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?

• 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?
– YCor
May 17, 2018 at 23:23
• @Ycor yes, that's true. May 18, 2018 at 9:37
• 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).
– YCor
May 18, 2018 at 9:44