  In this paper we consider the equivariant 2+1 dimensional Einstein-wave map
system and show that if the target satisfies the so called Grillakis condition,
then global existence holds. In view of the fact that the 3+1 vacuum Einstein
equations with a spacelike translational Killing field reduce to a 2+1
dimensional Einstein-wave map system with target the hyperbolic plane, which in
particular satisfies the Grillakis condition, this work proves global existence
for the equivariant class of such spacetimes.
