Towards tempered anabelian behaviour of Berkovich annuli

TL;DR

This paper explores partial anabelian properties of Berkovich analytic annuli, showing that isomorphic tempered fundamental groups imply the annuli have similar lengths, with differences bounded by residual characteristic.

Contribution

It demonstrates that in Berkovich geometry, isomorphic tempered fundamental groups constrain the lengths of analytic annuli, revealing a tempered form of anabelian behaviour.

Findings

Lengths of isomorphic annuli are close, with differences bounded by residual characteristic.

The absolute difference in lengths is bounded above by a constant depending only on p.

Partial anabelian behaviour is established for Berkovich annuli.

Abstract

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 $p$.