On \'etale fundamental groups of formal fibres of $p$-adic curves

Abstract

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) \'etale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric \'etale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.