Somewhere, I don't remember where, I saw a beautiful 3D figure of part a CAT(0) simplicial complex. I am thinking and hoping that this was some finite piece of an affine building of type A2, presumably in characteristic 2. But I'm very frustrated now that I just can't remember exactly what I saw or where I saw it. It was something like part of the neighborhood of radius 1 or radius 2 of a vertex, with enough simplices removed so that the rest fits in $\mathbb{R}^3$. It looked like Mathematica graphics, since (in my recollection) it had good colors to help show the 3-dimensional structure. I'm thinking that I saw it in the AMS Notices, but it could also have been in an AMS calendar or elsewhere. Does anyone remember seeing an image like this, and if so, where?

I'm asking because I'd like to have such an image in a paper that I'm working on, not necessarily the one that I saw but something similar.

I got some good answers to my question, both here and in private e-mail to Bill Casselman. But in the end I decided to make my own diagram (with the aid of TikZ and Python SAGE). Here it is.

alt text http://www.freeimagehosting.net/uploads/e38638d43e.png

For those who are interested in the TikZ and SAGE code, I combined them into one TeX document. I posted both the TeX source and its PDF output (from pdflatex) on my web page.

  • $\begingroup$ I saw many such pictures, and I know the particulars, but that surely won't help you, because they were all done on a blackboard in color chalk and erased afterwards! But on the positive side, I remember discussing software which produces such pictures with various geometric group theorists at Cornell Topology Fest (email me if you need the names). $\endgroup$ – Victor Protsak Jul 2 '10 at 2:20
  • $\begingroup$ Could you provide a link to your Tikz-Python code that generates the above figure? thanks. $\endgroup$ – Suvrit Nov 22 '10 at 11:08
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    $\begingroup$ Okay, I did that. - Greg $\endgroup$ – Greg Kuperberg Dec 26 '10 at 17:08

Check out http://dean.clas.uconn.edu/teitelbaum/colorout.gif

I made this image long ago for the cover of the AMS Notices.

See http://www.ams.org/notices/199510/teitelbaum.pdf

  • $\begingroup$ I rather like this example; it certainly serves the purpose that I had in mind. The figure that is in my mind is spiffier than this one, but on the other hand it could possibly be a recreated memory or a figure of something else. Actually, I don't just like your figure, I like your AMS article too. $\endgroup$ – Greg Kuperberg Jul 2 '10 at 17:05
  • $\begingroup$ Peter Schneider had a student who drew a picture like this for an expository article that he wrote, but I'm not sure where it is. $\endgroup$ – Jeremy Teitelbaum Jul 2 '10 at 22:21

A picture similar to the one you've made can be found in Garrett's book on Buildings and classical groups.



This may not be precisely what you ask for, but my colleague Paul Gunnells wrote a nice article for the May 2006 AMS Notices with some color pictures involving Kazhdan-Lusztig cells in small rank Coxeter groups. The link is http://www.ams.org/notices/200605/fea-gunnells.pdf

I have had some conversations with him about cells and have gotten some insight from the pictures he generated using graphics software (which probably wasn't Mathematica but also doesn't run on his current computing hardware). This has just been a sporadic interest of his, though he has an ongoing collaboration with Belolipetsky mostly involving cells in Coxeter groups which are not finite or affine.

Of course this kind of Coxeter group information is only one part of the building construction, but a crucial one to visualize.

ADDED: To follow up Greg's comment, there have been other items in the Notices dealing directly with Bruhat-Tits buildings. An article by K.S. Brown in the November 2002 issue has a nice $A_3$ picture: http://www.ams.org/notices/200210/what-is.pdf

(Most such articles and graphics have been influenced by the editorial work of Bill Casselman at UBC, who would be the best source of technical information.)

  • $\begingroup$ These are vaguely similar pictures, but they aren't branched. $\endgroup$ – Greg Kuperberg Jul 1 '10 at 22:56
  • $\begingroup$ True. While the alcoves and apartments in 3 dimensions can be visualized to some extent, the next step is to glue apartments along alcove walls, etc. It gets quite hard to picture buildings in enough detail. $\endgroup$ – Jim Humphreys Jul 1 '10 at 23:46

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