  We show that there are 2 equivalent first order descriptions of 2+1 gravity
with non-zero cosmological constant. One is the well-known spacetime
description and the other is in terms of evolving conformal geometry. The key
tool that links these pictures is Cartan geometry, a generalization of
Riemannian geometry that allows for geometries locally modelled off arbitrary
homogeneous spaces. The two different interpretations suggest two distinct
phase space reductions. The spacetime picture leads to the 2+1 formulation of
General Relativity due to Arnowitt, Deser, and Misner while the conformal
picture leads to Shape Dynamics. Cartan geometry thus provides an alternative
to symmetry trading for explaining the equivalence of General Relativity and
Shape Dynamics.
