1

The short answer:

You can't think about this building as embedded in $\mathbb{R}^3$. The building is really a simplicial complex where the chambers are simplices, so their interiors do not intersect. The visualizations below just aren't capable of showing this.

(The original question:

I am reading the presentation by Guyan Robertson pictured below, and I have seen this same claim in other sources, but I don't see how it is true: namely, that this "chamber complex" on which $SL_3(\mathbb{Q}_p)$ acts is a simplicial complex. This seems like a doubtful claim because it breaks this requirement of a simplicial complex: "Every pair of distinct simplices have disjoint interiors." There is no way that the simplices have disjoint interiors in this "building" that is constructed below.

When I try to draw the space (and even when he drew it on the last page below), the simplices (triangles) are intersecting each other's interiors. The simplex on the bottom of the diagram, for instance, intersects 5 other triangles through their interiors.

Could someone please explain to me how it is possible that the resulting structure is actually a simplicial complex?

The first option I can think of is that the intersections break each chamber into lots of smaller simplices. But the presentation says that chambers are maximal simplices, so this cannot be the answer. The chambers are simplices of codimension 2.

Then how does it make sense that the simplices are intersecting each other's interiors? Is the problem that I'm thinking in terms of 3-D our three-dimensional world that would force these simplices to intersect through their interiors? Is it possible to say "pretend like these intersections of interiors just don't exist, and only the edge intersections exist"?

But if we say "pretend those intersections aren't really happening", is this still a simplicial complex? Are supposed to somehow use that assumption when computing the homology of the space, for instance?

So really, an eqivalent question would be: What are the 1-chains on the ball of radius 1 in the picture below? Is a 1-chain allowed to travel freely along a face of a chamber, magically going through all the chambers seemingly intersecting it? (This is my main question.)

p.s. Keep in mind that every triangle below is filled by a disk. The whole space is contractible after all.

Also, I noticed one person here named @CheerfulParsnip said, "One needs to distinguish between an abstract simplicial complex and its geometric realization." So is the idea to think of the complex as something that does not actually exist geometrically, at least in physical-world geometry?

enter image description here

Your help would be so appreciated since this is really frustrating me!

p.p.s. Here is a better picture of the 2-simplices from https://buildings.gallery. In this diagram, the simplices clearly intersect each other's interiors, but are we just saying "pretend they don't" so that this will be a simplicial complex? Or do they appear to intersect in 3-D animations, but if we were thinking in higher dimensions, the interiors would not actually intersect?)

enter image description here

June in Juneau
  • 185
  • 1
    So far only vertex and edge data has been given. Is that right? This space is to be a $1$ dimensional complex i.e. a graph? – FShrike Jan 13 '24 at 12:42
  • @FShrike Thanks for coming! It is a 2-dimensional complex. Each triangle is filled. And it is a Euclidean building, which is contractible. Here is a better image of the 2-simplices: https://www.researchgate.net/figure/The-affine-building-of-typeAtype-typeA-2-connected-to-the-group-SL-3-Q-2-up-to_fig2_346302395 – June in Juneau Jan 13 '24 at 12:53
  • 1
    Well, if each triangle is filled, we should be given data on how exactly they are filled. To specify a simplicial set/complex you need to actually give the $2$-simplices – FShrike Jan 13 '24 at 13:03
  • @Fshrike Each chamber is equal to an element of the group multiplied by the initial chamber. I added the better picture to the question. The treatments of chamber complexes define them this way. (Even though chambers are simplices, and chamber complexes are simplicial complexes.) – June in Juneau Jan 13 '24 at 13:10
  • 1
    Sorry, I just feel like there are many pictures but it's missing precise mathematical context. I'm sure it's well-known what exactly $\Delta$ is defined to be, but I don't know it.. – FShrike Jan 13 '24 at 13:14
  • 1
    @FShrike Hopefully that is why I'm not understanding these papers. I will keep looking for more basic sources that give a precise definition of the 2-simplices, but I hope someone here can help clarify this because I haven't been able to find this so far. – June in Juneau Jan 13 '24 at 13:48
  • 1
    Did you check Ronan's "Buildings"? – Moishe Kohan Jan 13 '24 at 14:22
  • 1
    I am guessing that the intersections are just artifacts of the drawing in 3D. Also, a standard way of promoting a 1-complex to a simplicial complex is to fill in all simplices whose 1 skeleton is already in the complex. (Google "full simplicial complex.") – Cheerful Parsnip Jan 13 '24 at 14:45
  • @MoisheKohan Yes, I have it! I understand the action of the building on the space, which he describes really well. And Ronan refers to the chambers as 2-simplices. He also has this same building as an example. This leads me to believe the assumption is that the chambers' interiors don't actually intersect, even though this is pphysically impossible to visualize. That is, a chain or cycle can lie on a chamber with no issue. – June in Juneau Jan 13 '24 at 14:47
  • 1
    Does the book include this embedding of the simplicial complex into $\Bbb{R}^3$ that you are drawing/displaying? Does it include an embedding into some Euclidean space at all? I'm not used to simplicial complexes coming equipped with Euclidean embeddings. – Servaes Jan 13 '24 at 14:47
  • @CheerfulParsnip Thank you! I also was betting it was an artifact of the attempt to visualize the space, and now I'm sure that must be it. And googling "full simplicial complex" really helps since I'm finding papers that jump to homology groups of these buildings, which is my main question in the first place. – June in Juneau Jan 13 '24 at 14:55
  • @Servaes I have several books on buildings, and none of them try to embed the simplicial complex into $\mathbb{R}^3$, although many of them draw a 3-D diagram of some parts of the space. Now I'm seeing that these drawings are really just schematics because as you point out, there does not exist an embedding of the complex into $\mathbb{R}^3$! Each apartment is a copy of $\mathbb{R}^2$, so an embedding of some of the apartments into $\mathbb{R}^3$ is possible, but not the whole building, which I think is the answer to my question. – June in Juneau Jan 13 '24 at 15:01
  • 1
    Chamber's interiors must be disjoint. You probably just have a wrong picture. – Moishe Kohan Jan 13 '24 at 15:09
  • @MoisheKohan Thank you! Yes, that makes perfect sense. A bunch of papers and books provide those pictures, but I think the key is to not take them too seriously and focus on the group instead. – June in Juneau Jan 13 '24 at 15:10

0 Answers0