# Reference for the geometry of horospheres

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, DOI: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.

• I would write "Applying the comparison for triangle one which vertex running to infinity, we get ..." Commented May 31, 2010 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... Commented Jun 1, 2010 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. Commented Jun 7, 2010 at 12:34

Try Geometry of horospheres by Heintze and Im Hof.

• Thank you for the reference. In the meantime, a student of mine has written down a detailed proof using comparison theorems and looking at how one can pass to the limit when one vertex tends to infinity. This approach has the advantage that it works is general for CAT(-1) spaces. Commented Jun 15, 2010 at 18:07
• Heintze-Im Hof's paper certainly treats similar issues but does it contain an answer to your specific question? Where exactly? You may find relevant the paper of Bowditch "Geometric finiteness with variable negative curvature" where in section 2 he discusses similar issues, but again not your specific question. Commented Jun 15, 2010 at 18:51
• It seems that the link intlpress.com/JDG/archive/1977/12-4-481.pdf is not working (at least not for me). Here are some other links: projecteuclid.org/euclid.jdg/1214434219 dx.doi.org/10.4310/jdg/1214434219 mathscinet.ams.org/mathscinet-getitem?mr=512919 Commented Dec 1, 2017 at 13:56

I think you can find this in Chapter II.8 of Bridson and Haefliger.

• I gave a look therebut I didn't find the required estimate. It seems to me that in II.8 Bridson and Haefliger treat the general case of CAT(0) metric spaces, while I would need a result about CAT(-1) metric spaces. Commented May 31, 2010 at 16:57
• I thought it followed from what they do there, possibly throwing in strict convexity of the distance function. Perhaps I'm mistaken. Commented May 31, 2010 at 17:54

I am now also reading Farb´s article your cited, and I would like to propose an answer of mine for your question and help that we can discuss it.

Let $\gamma$ be a geodesic such that $d(γ(t),H)≥k>0$ for every $t∈[0,1]$ where H is a horosphere. Let $p = \gamma(0), q = \gamma(1)$, then length of $\gamma$,that is $L(\gamma)$, equals to $d(p,q)$. Let $p^\prime, q^\prime$ be the projection points of p,q respectively on horosphere. Suppose that $d(p,p^\prime)=k, d(q,q^\prime) \ge k$. Then choose a point $m$ in geodesic connecting $q,q^\prime$ so that $d(q^\prime,m)=k$ and connect $m$ to p by geodesic by curve $\delta$ which constructed by push geodesic $\gamma$ along geodesic with distance k.

By Proposition 4.1 in Farb´sRelatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840, $d(p^\prime,q^\prime) \le L(\delta ) \times e^{-ak}$. On the other hand, $L(\delta) \le L(\gamma)$ which can prove use the definition of the length and the fact that $\gamma$ is a geodesic.(I don´t know whether $\delta$ in this case is a geodesic, but it not affect the result).