Timeline for What spaces have well known horofunctions?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2015 at 13:02 | comment | added | Dylan Thurston | I think it's worth saying explicitly that Busemann functions are all horofunctions, but not necessarily vice versa. Teichmüller space is one counterexample to the converse. | |
| Oct 15, 2010 at 2:43 | vote | accept | Pablo Lessa | ||
| Sep 26, 2010 at 4:08 | answer | added | Ian Agol | timeline score: 2 | |
| Sep 25, 2010 at 9:54 | history | edited | Pablo Lessa | CC BY-SA 2.5 |
added tag reference-request
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| Sep 24, 2010 at 23:25 | comment | added | Igor Belegradek | Understanding Busemann functions involves understanding rays, which I think is a difficult task even for surfaces of revolution in $\mathbb R^3$. | |
| Sep 24, 2010 at 23:18 | answer | added | R W | timeline score: 5 | |
| Sep 24, 2010 at 20:02 | comment | added | Pablo Lessa | Well, no. Some discrete spaces such as infinite trees have plenty of horofunctions (e.g. one for each path to infinity). Horofunctions are interesting for example for Cayley graphs of finitely generated groups (with the so called "word metric"). The intuition is that as long as there are different ways of going "to infinity" there are horofunctions. However I was thinking more about Riemannian manifolds (even so, interesting examples of metric spaces are welcome as well). | |
| Sep 24, 2010 at 19:54 | answer | added | valeri | timeline score: 2 | |
| Sep 24, 2010 at 19:17 | comment | added | user47274 | and yet another silly question: does the same conclusion of my previous comment hold for discrete spaces (I'm thinking of graphs, trees actually)? | |
| Sep 24, 2010 at 19:16 | comment | added | Pablo Lessa | @Michele: You're correct, no horofunctions if the space is compact (the idea is that adding the horofunction one obtains a nice compactification of $X$ by "directions to infinity"). | |
| Sep 24, 2010 at 18:54 | comment | added | user47274 | Hi Pablo! Just to see if I've got something.. If X is compact there are no horofunctions, are there? | |
| Sep 24, 2010 at 18:49 | comment | added | Joseph O'Rourke | Just to supplement your posting, the horofunction definition (first?) appears in Gromov's 1980 paper, "Hyperbolic manifolds, groups and actions." ihes.fr/~gromov/PDF/hyperbolic_manifold.pdf | |
| Sep 24, 2010 at 18:32 | history | asked | Pablo Lessa | CC BY-SA 2.5 |