# Visibility spaces and Gromov hyperbolicity

I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic manifolds? Or even an implication. By visibility I mean:

A Riemannian manifold $(M,g)$ with riemannian distance $d$ is said to be a visibility manifold if given $p\in M$ and $\varepsilon>0$ there exists $r=r(p,\varepsilon)>0$ with this property: if $\sigma:[a,b]\longrightarrow M$ is a geodesic segment such that $d(p,\sigma)\geqslant r$, then $\sphericalangle_p(\sigma(a),\sigma(b))\leqslant\varepsilon$, where $\sphericalangle_p(\sigma(a),\sigma(b))$ is the angle between the tangent vectors of the minimizing geodesics joining $p$ to $\sigma(a)$ and to $\sigma(b)$.

The manifold is said to be uniform visibility manifold if $r$ doesn't depend on $p$.

I already know, based on the work of Morse, that the $\delta$-hyperbolic condition implies some sort of shadowind property of quasi-geodesics. Also, based on the work of Bonk, we see that the shadowing property implies $\delta$-hyperbolicity (Bonk, M.: Quasi-geodesic segments and Gromov Hyperbolic Spaces). I'm trying to understand the relation (if there exists some) between visibility, Gromov hyperbolicity and the shadowing property.

Thanks in advance.

• What is your definition of uniform visibility? – Misha Jun 24 '14 at 16:41
• I'll put this on the question for the sake of completeness. Thanks – matgaio Jun 24 '14 at 16:57
• Bridson-Haefliger's textbook has an extensive discussion of these matters. In particular, for proper CAT(0) spaces hyperbolicity and uniform visibility are equivalent (Proposition II.H.1.4), and in a proper, geodesic, hyperbolic space any two points at infinity are joined by a geodesic (Lemma III.H.3.2). – Igor Belegradek Jun 24 '14 at 22:19

## 2 Answers

For Cartan-Hadamard manifolds the equivalence of uniform visibility and Gromov hyperbolicity is proved in the following paper by Kaimanovich, Theorem 2.9.

• @R W Do you know by chance if there exists some extension of this result to manifolds without conjugate points? – matgaio Jun 24 '14 at 19:13
• @matgaio No idea - one has to check the argument - it might work in this situation as well. – R W Jun 24 '14 at 19:35
• @R W thanks a lot. It was really helpful. – matgaio Jun 24 '14 at 19:36

The visibility property is not only true for hyperbolic groups. A. Karlsson proved that it is true on the Floyd boundary (obtained as a completion of the Cayley graph equipped with a rescaled metric) of a finitely generated group (Comm. Algebra, 31, (2003), 5361--5376. lemma 1). it states that for every epsilon >0 there exists r=r(epsilon) such that every word geodesic outside of the ball of radius r has the Floyd distance between its endpoints less than epsilon. This is a very useful and less rigid condition (than a hyperbolicity). In particular it is also true for rel. hyperbolic groups on their Bowditch (relative) graph compatified with the Bowditch boundary. You can replace Floyd distance by topological entourages on the compactified space. Namely for every entourage on the compactified space there is a finite number of edges of the graph st if the pair of endpoints of the geodesic does not belong to the entourage then the geodesic passes through one of those edges. One needs to assume that the action of the group is rel. hyperbolic on the compactum or on the graph in any reasonable definition (see eg V.~Gerasimov, GAFA {\bf 19} (2009) (proposition 3.5.1). This "Karlsson lemma" remains valid for quasi-geodesics and even for alpha-geodesics where alpha is a reasonable distortion function (J. Eur. Math. Soc. {\bf 15} (2013), no. 6, 2115-2137, lemma 5.1) Best Lenya