Poincaré disk model is defined in a open disc, and the boundary of the disc represent something infinitely distant. But what the "something" exactly is?
How to topologically or geometrically extend the open disc in Poincaré disk model into a closed disc? Intuitively I think it is related with compactification of $H^2$. Is this a correct idea?
P.S. As some friend pointed to me, this should be related with conformal infinite or conformal boundary, and conformal compactification, ref https://arxiv.org/abs/1008.4703 but is that the whole answer?
The boundary of the Poincaré disk is most definitely not a topological object.
It is inherently a geometric object, and it is described here.
Here is a picture of single equivalence class of rays in the Poincare disc model, i.e. all lines in the Poincare disc sharing a common point at infinity.