Reference for the geometry of horospheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:10:47Z http://mathoverflow.net/feeds/question/26582 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26582/reference-for-the-geometry-of-horospheres Reference for the geometry of horospheres Roberto Frigerio 2010-05-31T14:26:59Z 2010-06-15T02:17:39Z <p>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).</p> <p>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 $e^{-\alpha k} L(\gamma)$.</p> <p>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.</p> http://mathoverflow.net/questions/26582/reference-for-the-geometry-of-horospheres/26592#26592 Answer by Matthew Stover for Reference for the geometry of horospheres Matthew Stover 2010-05-31T15:20:10Z 2010-05-31T15:20:10Z <p>I think you can find this in Chapter II.8 of Bridson and Haefliger.</p> http://mathoverflow.net/questions/26582/reference-for-the-geometry-of-horospheres/28194#28194 Answer by Igor Belegradek for Reference for the geometry of horospheres Igor Belegradek 2010-06-15T02:17:39Z 2010-06-15T02:17:39Z <p>Try <a href="http://www.google.com/url?sa=t&amp;source=web&amp;cd=4&amp;ved=0CCEQFjAD&amp;url=http%3A%2F%2Fwww.intlpress.com%2FJDG%2Farchive%2F1977%2F12-4-481.pdf&amp;ei=8-EWTOknwYHyBp6y3PMI&amp;usg=AFQjCNFd1SZWgJqlUWCA-8lYqLHdWUtP3w" rel="nofollow">Geometry of horospheres</a> by Heintze and Im Hof.</p>