Noncommutative two-dimensional gravities

We give formulations of noncommutative two-dimensional gravities in terms of noncommutative gauge theories. We survey their classical solutions and show that solutions of the corresponding commutative theories continue to be solutions in the noncommutative theories as well. We argue that the existence of ‘twisted’ diffeomorphisms, recently introduced in Aschieri et al (2005 Class. Quantum Grav. 22 3511), is crucial for this conclusion.