I am looking for a reference to a proof of the following well-know fact (cited for example by
B.Farb in ``Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840); [MR1650094](https://mathscinet.ams.org/mathscinet-getitem?mr=1650094), 
[DOI:10.1007/s000390050075](https://doi.org/10.1007/s000390050075).


Suppose $X$ is the universal covering of a negatively curved Riemannian manifold, let $O$ be an open horoball in $X$ and let $H=\partial O$ be the horospherical boundary of $O$. 
Also suppose that $\gamma\colon [0,1]\to X\setminus O$ is a rectifiable path such that $d(\gamma (t), H)\geq k>0$ for every $t\in [0,1]$, and let $\pi\colon X\setminus O\to H$
the (well-defined) nearest-point projection. Then, there exists $\alpha>0$ (only depending on the curvature of $X$) such that the length $L(\pi\circ\gamma)$ of $\pi\circ\gamma$ is bounded above by $e^{-\alpha k} L(\gamma)$.


Of course, this fact can be reduced to the computation of the Lipschitz constant of the projection of a horosphere onto another horosphere having the same basepoint.
When $X$ is the real hyperbolic $n$-space, such a computation is very easy, and it is likely that the variable curvature case can be reduced to the hyperbolic case via some comparison theorem. However, I was wondering if there is some standard reference I could rely on.



Reference for the geometry of horospheres
Try [Geometry of horospheres](http://www.google.com/url?sa=t&source=web&cd=4&ved=0CCEQFjAD&url=http%3A%2F%2Fwww.intlpress.com%2FJDG%2Farchive%2F1977%2F12-4-481.pdf&ei=8-EWTOknwYHyBp6y3PMI&usg=AFQjCNFd1SZWgJqlUWCA-8lYqLHdWUtP3w) by
Heintze and Im Hof.


