AdS/CFT UotC*

After praising Susskind's expository skills I find that he leaves me mystified on a crucial point. Maldacena conjectured a correspondence between Anti-de Sitterk space and a conformal field theory on its "boundary." Anti-de Sitter space, as far as I can tell, is a space with Lobachevskian spatial section - a space whose spatial part has constant negative scalar curvature. Susskind illustrates his AdS with Escher's picture of angels and demons on a Poincare disk. (I seem to recall Maldacena doing the same in a Scientific American story).

Poincare had noticed (a hundred years or so ago) that he could conformally map Lobachevsky's space on to a unit disk in the complex plane. The infinite Lobachevskian space is mapped onto a finite disk by a transformation that shrinks everything as it approaches the boundary.

The problem in my mind is that Susskind speaks of the boundary of the model disk as a real boundary at a finite distance, and treats the apparent shrinking of the far away angels, demons, or galaxies as real. What's up with that? What does it even mean to have a holographic image of your space at infinity? There must be a missing piece of the mental machinery that I can't see.
*Anti-de Sitter/Conformal Field Theory Unclear on the Concept

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