  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.
