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I'm study the Bruhat-Tits building of $PGL_n(K)$, where $K$ is a complete local field. In particular, im tried to understand the geometric properties of the apartment in special case $n=4$, which are tilings of Euclidean space $\mathbb{R}^{3}$. In the case $n=3$, the tiling (apartments) of the plane is generated by equilateral triangles. But in the case $n=4$ the tiling cannot be generated by regular tetrahedrons. So, which are the "good" tetrahedrons that tiling the space?

Im guess that the tetrahedrons have dihedral angles $\frac{\pi}{2}$, $\frac{\pi}{3}$. The angles are obtained by the Coxeter matrix $(m_{i,j})$ related with the Weyl group $W$ generated by reflections on the faces of fundamental tetrahedron, but im not sure of that.

Thanks.

P. Gudu
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  • Hello. While you're trying to understand the building of $PGL_4$, I'm starting out on the building of $PGL_3$. Maybe you can help out in that simpler case? https://math.stackexchange.com/questions/2982754/bruhat-tits-building-of-pgl-3-what-does-it-look-like – BharatRam Nov 03 '18 at 11:56

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