On triples of ideal chambers in $A_2$-buildings

Abstract

We investigate the geometry in a real Euclidean building $X$ of type $A_2$ of some simple configurations in the associated projective plane at infinity $\mathbb{P}$, seen as ideal configurations in $X$, and relate it with the projective invariants (from the cross ratio on $\mathbb{P}$). In particular we establish a geometric classification of generic triples of ideal chambers of $X$ and relate it with the triple ratio of triples of flags.