# Reference for the geometry of horospheres

Dear all, 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).

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.

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I would write "Applying the comparison for triangle one which vertex running to infinity, we get ..." –  Anton Petrunin May 31 '10 at 18:15
Yes, probably it is not too difficult to make such an argument work. Anyway, a little issue arises since one egde of the comparison triangles involved is not geodesic, but lies on a horosphere... –  Roberto Frigerio Jun 1 '10 at 7:50
Dear Anton, on second thoughts I think your approach can easily lead to a solution. Even if the edge staying far from the infinity is not geodesic, one can approximate $\gamma$ and $\pi\circ\gamma$ with suitable polygonal'' paths approximating the length, then use your argument on the small segments, and finally put the estimates together. –  Roberto Frigerio Jun 7 '10 at 12:34