  We study the embedded Calabi-Yau problem for complete embedded constant mean
curvature surfaces of finite topology or of positive injectivity radius in a
simply-connected three-dimensional Lie group X endowed with a left-invariant
Riemannian metric. We first prove a half-space theorem for constant mean
curvature surfaces. This half-space theorem applies to certain properly
immersed constant mean curvature surfaces of X contained in the complements of
normal R^2 subgroups F of X. In the case X is a unimodular Lie group, our
results imply that every minimal surface in X-F that is properly immersed in X
is a left translate of F and that every complete embedded minimal surface of
finite topology or of positive injectivity radius in X-F is also a left
translate of F.
