  We discuss negatively curved homogeneous spaces admitting a simply transitive
group of isometries, or equivalently, negatively curved left-invariant metrics
on Lie groups. Negatively curved spaces have a remarkably rich and diverse
structure and are interesting from both a mathematical and a physical
perspective. As well as giving general criteria for having left-invariant
metrics with negative Ricci curvature scalar, we also consider special cases,
like Einstein spaces and Ricci nilsolitons. We point out the relevance these
spaces play in some higher-dimensional theories of gravity. In particular, we
show that the Ricci nilsolitons are Riemannian solutions to certain
higher-curvature gravity theories.
