My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (*Nonspherical spheres*).


$[\ldots]$


Let $\mathcal{P} = \{P\_1, \ldots, P\_n\}$ be a minimal parabolic system for a group $G$, $B=P\_1 \cap \ldots \cap P\_n$ the Borel subgroup. A subgroup $N$ of $G$ is called a Weyl group for $\mathcal{P}$ iff


1) $N= \langle x\_1, \ldots, x\_n \rangle, x\_i \in P\_i-B, x\_i^2 \in B$.


2) $B \cap N$ is a normal subgroup of $N$.


3) $N \cap P\_i = (B \cap N) \langle x\_i \rangle , i = 1, \ldots, n$.


$[\ldots]$


If additionally we have


(A) $G = BNB$ and


(B) $BgBhB \subset (BgB) \cup (BghB)$ for all $g,h \in N$


then we have a $BN$-pair.
Geometrically, the Weyl group $N$ is the stabilizer of an apartment $\Delta$ of the geometry $\Gamma$ defined from $\mathcal{P}$, and $B$ is the stabilizer of a chamber of $\Delta$. I am trying to get a clear geometric view of these objects.


Here is my question. What could be a geometric interpretation of condition (B)?



Geometric interpretation of $BN$-pairs
Note: I'm using the general definition of BN-pair, which is weaker than the condition (B) given above.


It's a triangle inequality. One way to think about BN-pair is that they give a sort of combinatorial distance function on $G/B$. Given two cosets $g\_1B$ and $g\_2B$, you look at the product $Bg\_1^{-1}g\_2 B\in B\backslash G/B\cong N/(N\cap B):= W$ and think of that as the "distance" between them. To get a more numberish distance, you can let the length of an element of $W$ be the length of the shortest product of $x\_i$'s which gives it.


If I take two cosets $BgB$ and $BhB$, and expand those as $Bx\_{i\_1}\cdots x\_{i\_n}B$ and $Bx\_{j\_1}\cdots x\_{j\_m}B$, then $$BgB\cdot BhB\subset Bx\_{i\_1}\cdots x\_{i\_n}B\cdot Bx\_{j\_1}\cdots x\_{j\_m}B\subset Bx\_{i\_1}B\cdots Bx\_{i\_n}Bx\_{j\_1}B\cdots Bx\_{j\_m}B.$$ Applying (B) inductively, we see that the last term is in the union of certain double cosets which have length shorter than the sum of that of $BgB$ and $BhB$.


To apply this to the "distance function," note that $Bg\_1^{-1}g\_3B\subset Bg\_1^{-1}g\_2B\cdot Bg\_2^{-1}g\_3B$, so the length of the distance between $g\_1$ and $g\_3$ is less than the sum of the lengths for $g\_1$ to $g\_2$ and $g\_2$ to $g\_3$: the triangle inequality.


Of course, that just shows that (B) implies the triangle inequality, but it's easy to see that it's also a special case.


