Three-dimensional quantum geometry and black holes

We review some aspects of three-dimensional quantum gravity with emphasis in the ‘CFT → Geometry’ map that follows from the BrownHenneaux conformal algebra. The general solution to the classical equations of motion with anti-de Sitter boundary conditions is displayed. This solution is parametrized by two functions which become Virasoro operators after quantisation. A map from the space of states to the space of classical solutions is exhibited. Some recent proposals to understand the Bekenstein-Hawking entropy are reviewed in this context. The origin of the boundary degrees of freedom arising in 2+1 gravity is analysed in detail using the Hamiltonian Chern-Simons formalism.