Local constancy in families of non-abelian Galois representations

If S is a a scheme of finite type over a local field F, and X --> S is a pr oper smooth family, then to each rational point s is an element of S one ca n assign an extension of the absolute Galois group of F by the geometric fu ndamental group G of the fibre X-s. If F has uniformiser pi, and residue ch aracteristic p, we show that the corresponding extension of the absolute Ga lois group of S by the maximal prime to p quotient of G is locally constant in the pi-adic topology of S. We give a similar result in the case of non- proper families, and families over pi-adic analytic spaces. Mathematics Sub ject Classification (1991): 32P05, 14G20.