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Let $\mathbb{F}_{p}$ be a finite field of order $p$, and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. My question is the following:

Let $G$ be a closed pro-$p$ subgroup of ${\rm GL}_{n}(\mathbb{F}_{p}[[T]])$. Suppose that $G$ is topologically finitely generated and $G$ is torsion-free. Is it true that $G$ is solvable?

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  • $\begingroup$ I'd guess the answer is no at least for $n\ge 3$. Namely considering SO of a quadratic anisotropic form. This is a compact subgroup and I expect it to be virtually torsion-free (but has an abstract free subgroup). For $n=2$ an SU2-like subgroup might do the job, using that we can view $F_p((t))$ as quadratic extension of $F_p((t^2))$ to define an anisotropic Hermitian form. $\endgroup$
    – YCor
    Dec 10 at 5:33

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