Prime-to-$p$ \'etale fundamental groups of punctured projective lines over strictly Henselian fields

Let $K$ be the fraction field of a strictly Henselian DVR of characteristic $p\ge 0$ with algebraic closure $\bar{K}$, and let $\alpha_{1}, …, \alpha_{d} \in $\mathbb{P}${K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$p$ 'etale fundamental group of ${\mathbb{P}}_K^1\setminus \{{\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_d\}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_d$ be distinct power series in $\mathbb{C}[[x]]$ with the same intersection behavior as the ${\alpha }_i$’s, converging on an open disk centered at $0$, and choose a point $z_0\mathrm{\ne }0$ lying in this open disk. We compare the natural action of $\mathrm{G}\mathrm{a}\mathrm{l}(K)$ on the prime-to-$p$ 'etale fundamental group of ${\mathbb{P}}_{\bar{K}}\setminus \{{\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_d\}$ to the topological action of looping $z_0$ around the origin on the fundamental group of ${\mathbb{P}}_{\mathbb{C}}^1\setminus \{a_1(z_0),\mathrm{.}\mathrm{.}\mathrm{.},a_d(z_0)\}$. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-$p$ $G$-Galois cover of $$\mathbb{P}${\bar K}^1 \smallsetminus {\alpha_1,…,\alpha_d}$ satisfies that its field of moduli (as a $G$-Galois cover) has degree over $K$ dividing the exponent of $G / Z(G)$.