On a spacetime duality in 2+1 gravity

We consider 2+1 dimensional gravity with a cosmological constant, and explore a duality that exists between space-times that have the De Sitter group SO(3,1) as its local isometry group. In particular, the Lorentzian theory with a positive cosmological constant is dual to the Euclidean theory with a negative cosmological constant. We use this duality to construct a mapping between apparently unrelated space-times. More precisely, we exhibit a relation between the Euclidean BTZ family and some $T^2$-cosmological solutions, and between De-Sitter point particle space-times and the analytic continuations of Anti-De Sitter point particles. We discuss some possible applications for BH and AdS thermodynamics.