Timeline for Gromov's Hyperbolicity and Positive Cheeger Constant in Planar Graphs
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Mar 27, 2011 at 22:06 | vote | accept | ght | ||
| Mar 27, 2011 at 20:08 | comment | added | HJRW | @Gabriel, thanks for the reference. Regarding Joseph's comment, there are many definitions of $\delta$-hyperbolicity - indeed, the definition that you give is, I believe, due to Rips, not Gromov. I think the heart of Joseph's claim is that the converse of the proposition you state is also true. | |
| Mar 27, 2011 at 19:36 | comment | added | ght | @HW: See the paper for a more detailed discussion: "Cheeger Isoperimetric Constants of Gromov-Hyperbolic Spaces with Quasi-Poles" by J. Cao, published in Communications in Contemporary Mathematics, Vol. 2, No. 4, pp. 511–533, 2000. @Joseph: The definition of Gromov's hyperbolic graph is that there exists $\delta>0$ such that all the geodesic triangles in the graph are $\delta$ thin. This means that any side of a triangle is in the $\delta$ neighborhood of the other two. As you said this implies that every two geodesic rays starting in a point diverge exponentially | |
| Mar 27, 2011 at 19:24 | answer | added | Sam Nead | timeline score: 9 | |
| Mar 27, 2011 at 19:20 | comment | added | Joseph O'Rourke | I think this is one definition of a Gromov hyperbolic graph: any two of its geodesics are either parallel or diverge exponentially. Maybe Gabriel can elaborate further ... | |
| Mar 27, 2011 at 18:10 | comment | added | HJRW | Could you explain how 'the concepts of Gromov's hyperbolicity and strictly positive Cheeger constant are related'? | |
| Mar 27, 2011 at 17:46 | history | asked | ght | CC BY-SA 2.5 |