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?
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?)
