We study the buildings in which parallelism of residues is an equivalence relation. If the building admits a group action, we describe how parallel residues are related to residues with equal stabilizers. This permits to retrieve the fact that in a Coxeter group or in a graph product, intersections of parabolic subgroups are parabolic.
"Transitive parallelism of residues in buildings." Bull. Belg. Math. Soc. Simon Stevin 23 (4) 625 - 640, november 2016. https://doi.org/10.36045/bbms/1480993592