I am trying to study (finite) spherical buildings from a very combinatorial point of view : Every rank 3 spherical building is a finite simplicial complex of dimension 3, so one can define its density as the ratio #triangles/#vertices, expressed as a function of the number $n$ of vertices. Maybe it is the wrong name for a very classical notion but I did not find it anywhere. One of the first distinctions in buildings is the concept of thick buildings. In a way, this density measures the *thickness* of a given building, but I never saw this notion.


I am wondering what is the density of the spherical buildings, and which one are the densest ?


Starting from the classification of spherical buildings, it is enough to compute the density of the buildings of type $A\_3$ and $C\_3$. I know how to do it for $A\_3$, since the building in this case is the flag complex of a projective space on $\mathbb{F}\_q^4$, and it seems to give $\frac{[4]\_q!}{\binom{4}{2}\_q}=O(q^2)=O(\sqrt{n})$ where $n$ is the number of vertices, with the usual [q-analog](http://en.wikipedia.org/wiki/Q-analog) notations.


But $C\_3$ looks scary to me, as there are multiple possible geometries, and I don't know of a systematic way to extract the (simple) combinatorial information out of the groups. Is this something well known, or is there a simple way to do this ?



Density/Thickness of rank 3 spherical buildings
While I have never seen this notion of "density" of a finite building before, let's see what we can do...


A thick finite $C\_3$ building corresponds to a thick finite rank 3 polar space. These are well-known, they are $W(5,q)$, $Q(6,q)$, $Q^-(7,q)$, $H(5,q^2)$ and $H(6,q^2)$, corresponding to the groups $Sp(6,q)$, $O(7,q)$, $O^-(8,q)$, $U(6,q)$ and $U(7,q)$.


To compute the densities, I took a look at a table indicating the number of points contained in these spaces as well as the number of points collinear to a single fixed point (in my case, I looked at the book [Diagram Geometry](http://www.springer.com/mathematics/geometry/book/978-3-642-34452-7) by Buekenhout and Cohen, p. 485; there are other, older sources, but I already had that book sitting next to me... Google just showed me [another source here](http://www.maths.qmul.ac.uk/~pjc/pps/pps6.pdf)).


From this and the knowledge how many points are contained in a line and a plane (namely $q+1$ and $q^2+q+1$), it is a matter of simple combinatorics to count the total number of points, lines and planes, as well as chambers (= maximal flags = triples of a point, line and plane containing each other). Each point, line and plane corresponds to a vertex in the simplicial complex, while the chambers correspond to triangles. Thus the "density" as you defined it can be computed via the formula
$$ \frac{\#chambers}{\#points + \#lines + \#planes}.$$


Plugging in the numbers of points and perp sizes for all of the $C\_3$ buildings listed above, the "density" turns out to be in $O(q^3)$ in all cases (modulo mistakes on my side resp. in the small GAP program I wrote to compute all of this). So it is a bit larger than for projective spaces, where I confirmed it to be in $O(q^2)$.


