  Consider an ergodic unimodular random one-ended planar graph $\G$ of finite
expected degree. We prove that it has an isometry-invariant locally finite
embedding in the Euclidean plane if and only if it is invariantly amenable. By
"locally finite" we mean that any bounded open set intersects finitely many
embedded edges. In particular, there exist invariant embeddings in the
Euclidean plane for the Uniform Infinite Planar Triangulation and for the
critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking,
a unimodular embedding of $\G$ is one that is jointly unimodular with $\G$ when
viewed as a decoration. We show that $\G$ has a unimodular embedding in the
hyperbolic plane if it is invariantly nonamenable, and it has a unimodular
embedding in the Euclidean plane if and only if it is invariantly amenable.
Similar claims hold for representations by tilings instead of embeddings.
