  We identify an ambiguity in the Chern-Simons formulation of three-dimensional
gravity with negative cosmological constant that originates in an outer
automorphism of the Lie algebra sl(2,R). It has important consequences for the
stability of the theory in a space-time with boundary. We revisit the classical
equivalence of three-dimensional gravity with a boundary Liouville theory both
on and off the mass shell. Moreover, we provide further details on the quantum
equivalence, the gauge symmetry that renders the spectrum diagonal, as well as
the relation between asymptotically AdS3 metrics and polar boundary conditions.
We thus set the proposal that the Liouville conformal field theory serves as a
definition of a unitary theory of pure gravity in three dimensions with
negative cosmological constant on a more stable footing.
