#### Vol. 16, No. 2, 2022

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Tamely ramified covers of the projective line with alternating and symmetric monodromy

### Renee Bell, Jeremy Booher, William Y. Chen and Yuan Liu

Vol. 16 (2022), No. 2, 393–446
##### Abstract

Let $k$ be an algebraically closed field of characteristic $p$ and $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group ${S}_{n}$ or alternating group ${A}_{n}$ for infinitely many $n$. These covers come from the moduli spaces of elliptic curves with ${\mathrm{PSL}}_{2}\left({\mathbb{𝔽}}_{\ell }\right)$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder about Markoff triples modulo $\ell$.

##### Keywords
finite fields, tame fundamental group, Markoff triples, tamely ramified covers, characteristic $p$, covers of curves
##### Mathematical Subject Classification
Primary: 11G20, 14H30