Let $A \in \mathrm{GL}(n,\mathbb Z)$ be a torsion, I would like to prove that $\mathrm{order}(A)\leq K\exp (cn^{\alpha})$, with $0<\alpha <1$, for $n$ "large enough".
I know that if $\mathcal{H}(n)$ is the maximal finite order of an element of $\mathrm{GL}(n,\mathbb Z)$ then $$\lim_{n\to \infty} \frac{\ln \mathcal{H}(n)}{\sqrt{n\ln n}}= 1,$$ which is a stronger result than what I asked for, but I'm hoping for a shorter proof for my weaker result.