Transitive parallelism of residues in buildings

Abstract

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.

To appear in the Bulletin of the BMS. 14 pages, 1 figure. Partially rewritten after referees comments