This week is the third week since I started my PhD at Ghent University. The objective is to investigate properties of certain geometric objects called right-angled buildings and algebraic information on locally compact groups occurring as symmetry groups of these structures. I am still working my way into the definition of a (mathematical) building, which consists of chambers, arranged in apartments satisfying certain axioms, or obeying some group-valued “distance function” telling you how to move between chambers. Buildings were introduced by Belgian’ mathematician Jacques Tits.
It’s pretty vague, and maybe I’ll try to give a better introduction to the concept later, but it’s also really exciting to learn about these structures and to investigate their properties; they generalize lots of other geometric objects in a single beautiful framework.
In the picture you can see a standard example of a building, one of type A2. Quick overview: you see a graph with 14 vertices, being the points and lines of the Fano plane. We connect two vertices with an edge—and these edges will be the chambers of the building—whenever it connects a line with a point on that line. Hence, an edge can be seen as a flag (an incident point-line pair) of the Fano plane. Two flags can be incident in two different ways: they can have a common line, or a common point (or not be incident at all). Thus, the chambers can be adjacent in two different ways, and we have a rank 2 building. Finally, notice that the graph contains lots of cycles of length 6. These 6-cycles will be the apartments in the building, and you can check an important property or axiom: any two chambers (edges) lie in a common apartment (6-cycle).
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