I'm reading the book of Drutu and Kapovich "Lectures on geometric group theory". In the proof of Mostow rigidity theorem, they say that they can extend an $\rho$-equivariant function $f:\Omega\to\Omega'$ with $\Omega,\Omega'$ truncated hyperbolic spaces and it's easy from the follow fact:
The homomorphism $\rho$ sends maximal abelian subgroups of $\Gamma$ to maximal abelian subgroups of $\Gamma'$ . The stabilizers of peripheral horospheres are virtually $\mathbb{Z}^{n−1}$ . Therefore, $\rho$ sends stabilizers of peripheral horospheres to stabilizers of peripheral horospheres. From this, it is immediate that peripheral horospheres map uniformly close to peripheral horospheres.
I don't know how this fact define the extension and how it proofs that the extension is quasi-isometric. I'm following this book, theorem 22.32
https://www.math.ucdavis.edu/~kapovich/EPR/kapovich_drutu.pdf