By Riccardo Benedetti, Francesco Bonsante

The authors advance a canonical Wick rotation-rescaling conception in three-dimensional gravity. This contains: a simultaneous class: this indicates how maximal globally hyperbolic house occasions of arbitrary consistent curvature, which admit an entire Cauchy floor and canonical cosmological time, in addition to complicated projective buildings on arbitrary surfaces, are all assorted materializations of 'more basic' encoding constructions; Canonical geometric correlations: this indicates how house instances of alternative curvature, that proportion a related encoding constitution, are concerning one another through canonical rescalings, and the way they are often reworked through canonical Wick rotations in hyperbolic 3-manifolds, that hold the best asymptotic projective constitution. either Wick rotations and rescalings act alongside the canonical cosmological time and feature common rescaling features. those correlations are functorial with recognize to isomorphisms of the respective geometric different types

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Thus the X ˆ 1 can be pushed forward to X1 . In what follows we consider always metric on X X1 endowed with such a metric and we call it the Klein model of de Sitter space . Notice that it is an oriented spacetime (indeed it carries the orientation induced ˆ 1 → X1 are not by P3 ) but it is not time-oriented (automorphisms of the covering X time-orientation preserving). Since the automorphism group {±Id} is the center of the isometry group of O(3, 1), X1 is an isotropic Lorentz spacetime. The isometry group of X1 is O(3, 1)/± Id.

LS does not depend on the measure µ. In general this is not a sub-lamination, that is its support LS is not a closed subset of H. A leaf, l, is called weighted if there exists a transverse arc k such that k ∩ l is an atom of µk . By property (3) of the deﬁnition of transverse measure, if l is weighted then for every transverse arc k the intersection of k with l consists of atoms of µk whose masses are equal to a positive number A independent of k. We call this number the weight of l. The weighted part of λ is the union of all the weighted leaves.

We make this choice because in this way the causal structure on ∂X−1 is the “limit” of the causal structure on X−1 in the following sense. Suppose An to be a sequence in X−1 converging to A ∈ ∂X−1 , and suppose Xn ∈ TAn X−1 to be a sequence of timelike vectors converging to X ∈ TA ∂X−1 , then X is non-spacelike with respect to the causal structure of the boundary. Notice that oriented left (resp. right) leaves are homologous non-trivial simple cycles on ∂X−1 , so they determine non-trivial elements of H1 (∂X−1 ) that we denote by cL and cR .