Timeline for Angles and Busemann function in CAT(0)
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2021 at 15:01 | comment | added | AMath91 | Yes, that’s exactly what I need. I’ll ask a new question. Thanks! | |
| Jul 2, 2021 at 14:51 | comment | added | Stefan Witzel | Perhaps it's really time you asked a new question. How would $x$ and $y$ be contained in a common chamber in a non-discrete building? I think the statement that is still true (and implies the above in the discrete case) is that for a point $z \in X$ and a point $\xi \in \partial X$ there is an apartment containing both. (You recover the above by taking $z$ to be an interior point of the chamber.) I guess, what you want is an apartment that contains a germ from $x$ (toward y) as well as $\xi$ in the boundary and that could well be true, but I don't have a reference ready for that. | |
| S Jul 2, 2021 at 14:42 | vote | accept | AMath91 | ||
| S Jul 2, 2021 at 14:42 | vote | accept | AMath91 | ||
| S Jul 2, 2021 at 14:42 | |||||
| Jul 2, 2021 at 14:42 | vote | accept | AMath91 | ||
| S Jul 2, 2021 at 14:42 | |||||
| Jul 2, 2021 at 14:42 | comment | added | AMath91 | Do you know if that result holds for non-discrete buildings as well? I’m confused about which results of that book hold in general or only in the discrete case. | |
| Jul 2, 2021 at 14:40 | comment | added | Stefan Witzel | Theorem (1) on page 170 of Brown "Buildings". | |
| Jul 2, 2021 at 14:39 | answer | added | Stefan Witzel | timeline score: 1 | |
| Jul 2, 2021 at 14:35 | comment | added | AMath91 | Do you have a reference for this? I’m a novice at buildings and I haven’t found this result in the axioms. | |
| Jul 2, 2021 at 14:11 | comment | added | Stefan Witzel | yes given a chamber and a point at infinity you can find an apartment that contains the apartment and the point at infinity. This apartment being convex and containing $x$ as well as the endpoint of $\rho$, it will contain all of $\rho$. | |
| Jul 2, 2021 at 12:08 | comment | added | AMath91 | I wanted to use an argument like this, but I’m worried that this apartment will not contain the whole ray $\rho$. Assuming $x$ and $y$ are in the a common chamber, can I always find an apartment containing $y$, $x$ and $\rho$? | |
| Jul 2, 2021 at 12:03 | comment | added | AMath91 | In view of HJRW’s answer, is the statement true if I change obtuse with not acute (so I allow $\pi/2$)? | |
| Jul 2, 2021 at 10:47 | comment | added | Stefan Witzel | If "close" means that x and y lie in a common (closed) chamber then you can take an apartment that contains that chamber and contains xi in its boundary. Then you can use that what you want is a "known fact" for Euclidean spaces (specifically this apartment). | |
| Jul 2, 2021 at 10:16 | history | became hot network question | |||
| Jul 2, 2021 at 7:53 | answer | added | HJRW | timeline score: 5 | |
| Jul 1, 2021 at 22:55 | history | asked | AMath91 | CC BY-SA 4.0 |