Connectivity of chamber graphs of buildings and related complexes

TL;DR

This paper proves that the chamber graphs of finite buildings and related complexes are highly connected, specifically q-connected, which enhances understanding of their structural robustness.

Contribution

The paper establishes the q-connectivity of chamber graphs of finite buildings, Coxeter complexes, and geometric lattice order complexes, extending known regularity results.

Findings

Chamber graphs of finite buildings are q-connected.

Similar connectivity results hold for Coxeter complexes.

Order complexes of geometric lattices also exhibit q-connectivity.

Abstract

Let \Delta be a finite building (or, more generally, a thick spherical and locally finite building). The chamber graph G(\Delta), whose edges are the pairs of adjacent chambers in \Delta, is known to be q-regular for a certain number q=q(\Delta). Our main result is that G(\Delta) is q-connected in the sense of graph theory. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.