  The regular black hole solution arising as a spherically symmetric vacuum
solution of Born-Infeld gravity possesses an asymptotic interior structure
which is very well described by a four dimensional generalization of the
non-rotating BTZ metric. According to this picture no singularity exists, and
instead, infalling observers experience a constant curvature manifold as they
travel towards future null infinity. This is characterized by the BTZ event
horizon. The exterior structure of the black hole is also studied, and it is
shown that it corresponds to the Schwarzschild solution provided the black hole
mass is not too small. In this way, the regular black hole state can be seen as
a spacetime which connects two constant curvature asymptotic spaces, namely,
the flat Minkowski spacetime in the outside region, and the locally AdS
constant negative curvature one characterizing the BTZ-like asymptotic
interior.
