The horizon of the BTZ black hole

This paper is a sequel of "Solvable symmetric black hole in anti de Sitter spaces" [arXiv:math.DG/0510442]. In the latter, we described the BTZ black hole in every dimension by defining the singularity as the closed orbits of the Iwasawa subgroup of SO(2,n). In this article, we study the horizon of the black hole and we show that it is expressed as lateral classes of one point of the space. The computation is given in the four-dimensional case, but it makes no doubt that it can be generalized to any dimension. The main idea is to define an "inclusion map" from AdS_3 into AdS_4 and to show that all the relevant structure pass trough the inclusion. We prove, for example, that the inclusion of the three dimensional horizon into AdS_4 belongs to the four dimensional horizon. Then we deduce the expression of the horizon in AdS_4.