Intersections of Apartments

TL;DR

This paper characterizes the conditions under which the intersection of two apartments in a building can be any convex subcomplex, highlighting the importance of the complete system of apartments and the number of chambers per panel.

Contribution

It establishes a combinatorial characterization of apartment intersections in buildings, especially for lower-dimensional convex subcomplexes, under specific conditions.

Findings

Complete apartment systems allow any convex subcomplex as an intersection.

The four chambers per panel condition is necessary for certain convex subcomplexes.

Counter-examples exist for infinite convex subcomplexes in arbitrary systems.

Abstract

We show that, if a building is endowed with its complete system of apartments, and if each panel is contained in at least four chambers, then the intersection of two apartments can be any convex subcomplex contained in an apartment. This combinatorial result is particularly interesting for lower dimensional convex subcomplexes of apartments, where we definitely need the assumption on the four chambers per panel in the building. The corresponding statement is not true anymore for arbitrary systems of apartments, and counter-examples for infinite convex subcomplexes exist for any type of buildings. However, when we restrict to finite convex subcomplexes, the above remains true for arbitrary systems of apartments if and only if every finite subset of chambers of the standard Coxeter complex is contained in the convex hull of two chambers.