Birkhoff’s theorem for three-dimensional AdS gravity

All three-dimensional matter-free space-times with negative cosmological constant, compatible with cyclic symmetry, are identified. The only cyclic solutions are the $2+1$ (BTZ) black hole with $\mathrm{SO}\left(2\right)\mathrm{}\times R$ isometry, and the self-dual Coussaert-Henneaux space-times, with isometry groups $\mathrm{SO}\left(2\right)\mathrm{}\times \mathrm{SO}(2,1)\mathrm{}$ or $\mathrm{SO}\left(2\right)\mathrm{}\times \mathrm{SO}\left(2\right).$