This question is inspired by Upper bound on order of finite subgroups of GL_n(Z_p)?. It's showed that the supremum of orders of finite subgroups of ${\rm GL}_n(\mathbb{Z}_p)$ is finite and can be explicitly bounded, and there are arbitrarily large finite subgroups of ${\rm GL}_n(\mathbb{F}_q[[T]])$ for all prime powers $q$ and all $n\geq 2$. However, it's not hard to show that there is always a bound of the order of torsion elements in ${\rm GL}_n(\mathbb{F}_q[[T]])$, cf. Lemma 5.5. in A topological Tits alternative. My question is the following:

Let $n\geq 2$, $R=\mathbb{Z}_p$ or $R={\rm GL}_n(\mathbb{F}_p[[T]])$, and let $B(n,p)$ denote the maximum order of torsion elements in ${\rm GL}_n(R)$. What is $B(n,p)$? Can we get some estimates on it?

Any references is highly appreciated.

This question is inspired by Upper bound on order of finite subgroups of GL_n(Z_p)?. It's showed that the supremum of orders of finite subgroups of ${\rm GL}_n(\mathbb{Z}_p)$ is finite and can be explicitly bounded, and there are arbitrarily large finite subgroups of ${\rm GL}_n(\mathbb{F}_q[[T]])$ for all prime powers $q$ and all $n\geq 2$. However, it's not hard to show that there is always a bound of the order of torsion elements in ${\rm GL}_n(\mathbb{F}_q[[T]])$, cf. Lemma 5.5. in A topological Tits alternative. My question is the following:

Let $n\geq 2$, $R=\mathbb{Z}_p$ or $R=\mathbb{F}_p[[T]]$, and let $B(n,p)$ denote the maximum order of torsion elements in ${\rm GL}_n(R)$. What is $B(n,p)$? Can we get some estimates on it?

Any references is highly appreciated.