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Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not true that

(*) every two points in $\partial X$ can be connected by a geodesic in $X$.

However, one can perform the following trick, which allows one to reduce many questions to those spaces which actually satisfy (*).

Let $Y$ denote the metric ultrapower of $X$ with respect to a non-principal ultrafilter. Then $Y$ is also hyperbolic and there is a natural isometric embedding $X\to Y$, which extends to $\partial X\to \partial Y$. It is not difficult to show that any two points of $\partial X$ can be joined by a geodesic in $Y$.

My question is the following.

Are the details of this construction written in any paper or book?

This was a part of "hyperbolic folklore" for a long time and all proofs are straightforward, but I could not find any written detailed proof which can serve as a reference.

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  • $\begingroup$ I would just write a proof, it takes couple of lines. For a discussion of ultralimits, I usually refer to "Rigidity of quasi-..." by Klein and Leeb. $\endgroup$ May 26, 2018 at 0:07
  • $\begingroup$ I just want to give a proper credit in a survey paper. $\endgroup$
    – Denis Osin
    May 26, 2018 at 10:19
  • $\begingroup$ It doesn't look like it is in the GGT book by C. Druţu and M. Kapovich, although it does have discussion of ultralimits of metric spaces $\endgroup$
    – user35370
    May 26, 2018 at 19:23

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