Towards tempered anabelian behaviour of Berkovich annuli

This work brings to light some partial \emph{anabelian behaviours} of analytic annuli in the context of Berkovich geometry. More specifically, if $k$ is a valued non-archimedean complete field of mixed characteristic which is algebraically closed, and $\mathcal{C}_1$, $\mathcal{C}_2$ are two $k$-analytic annuli with isomorphic tempered fundamental group, we show that the lengths of $\mathcal{C}_1$ and $\mathcal{C}_2$ cannot be too far from each other. When they are finite, we show that the absolute value of their difference is bounded above with a bound depending only on the residual characteristic $p$.