An Oscillation Theorem for Groups of Isometries

Abstract.

This paper answers a question raised by Vladimir Pestov.

Let G be a non-trivial group of isometries of a complete, separable metric space. Let Ĝ be the semi-group arising from its completion with respect to the left uniformity. Then there is a uniformly continuous function from Ĝ to the unit interval which assumes both extreme points on any non-empty right ideal.