  According to the 't Hooft-Susskind holography, the black hole
entropy,$S_\mathrm{BH}$, is carried by the chaotic microscopic degrees of
freedom, which live in the near horizon region and have a Hilbert space of
states of finite dimension $d=\exp(S_\mathrm{BH})$. In previous work we have
proposed that the near horizon geometry, when the microscopic degrees of
freedom can be resolved, can be described by the AdS$_2[\mathbb{Z}_N]$
discrete, finite and random geometry, where $N\propto S_\mathrm{BH}$. It has
been constructed by purely arithmetic and group theoretical methods in order to
explain, in a direct way, the finiteness of the entropy, $S_\mathrm{BH}$. What
has been left as an open problem is how the smooth AdS$_2$ geometry can be
recovered, in the limit when $N\to\infty$. In the present article we solve this
problem, by showing that the discrete and finite AdS$_2[\mathbb{Z}_N]$ geometry
can be embedded in a family of finite geometries, AdS$_2^M[\mathbb{Z}_N]$,
where $M$ is another integer. This family can be constructed by an appropriate
toroidal compactification and discretization of the ambient $(2+1)$-dimensional
Minkowski space-time. In this construction $N$ and $M$ can be understood as
"infrared" and "ultraviolet" cutoffs respectively. The above construction
enables us to obtain the continuum limit of the AdS$_2^M[\mathbb{Z}_N]$
discrete and finite geometry, by taking both $N$ and $M$ to infinity in a
specific correlated way, following a reverse process: Firstly, by recovering
the continuous, toroidally compactified, AdS$_2[\mathbb{Z}_N]$ geometry by
removing the ultraviolet cutoff; secondly, by removing the infrared cutoff in a
specific decompactification limit, while keeping the radius of AdS$_2$ finite.
It is in this way that we recover the standard non-compact AdS$_2$ continuum
space-time. This method can be applied directly to higher-dimensional AdS
spacetimes.
