Reduction of the group of a (wildly ramified) Galois cover, and application to moduli.

Abstract. Let $R$ be a discrete valuation ring, $K=Frac(R)$ and $k$ the residue field. Let $G$ be a finite group whose order $n$ is a multiple of $p=car(k)$, assumed to be positive. If $X$ is a curve over $R$, a generically faithful action of $G$ can degenerate on the special fibre of $X$. This problem has nasty consequences for the study of moduli; we suggest an attempt to modify the action so as to obtain a "better" object than the pair $(X,G)$.