---
tag: 70Q4YVF322
kind: definition
name: Building
msc: ["20E42", "51E24"]
defines: 6BQJ50NEJ0
uses: [V8GZ0W1ZZ5, G6FND7X69J, 062D9KXJB1]
sources:
  - url: https://en.wikipedia.org/wiki/Building_(mathematics)
---

## Definition

Let $(W, S)$ be a [[Coxeter system|G6FND7X69J]].
For $J \subseteq S$, let $W_J$ be the subgroup of $W$ generated by $J$.
The Coxeter complex $\Sigma(W, S)$ is the [[simplicial complex|V8GZ0W1ZZ5]] whose simplices are the cosets $w W_J$ with $w \in W$ and $J \subseteq S$.
A simplex $w W_J$ is a face of $w' W_{J'}$ when $w W_J \supseteq w' W_{J'}$.
A *building* is a simplicial complex $\Delta$ together with a family of subcomplexes, called *apartments*, such that:

1. Each apartment is isomorphic to a Coxeter complex.
2. For any two simplices $A$ and $B$ of $\Delta$, there is an apartment that contains both.
3. If two apartments $\Sigma$ and $\Sigma'$ both contain the simplices $A$ and $B$, then some isomorphism $\Sigma \to \Sigma'$ fixes every vertex of $A$ and of $B$.

The maximal simplices are the *chambers*.
The building is *thick* if each face of codimension one of a chamber lies in at least three chambers.

## Remarks

Example: let $\Delta$ be a [[tree|062D9KXJB1]] in which every vertex has degree at least $3$, and let the apartments be the bi-infinite paths in $\Delta$.
Each such path is the Coxeter complex of the infinite dihedral group, a line divided into edges.
Clause 2 holds because the path between two edges extends at both ends to a bi-infinite path, since no vertex has degree $1$.
Clause 3 holds because two bi-infinite paths that contain $A$ and $B$ both contain the unique path between them, and a shift or reflection of lines matches them along it.
Each vertex lies in at least three edges, so the building is thick.

Non-example: a finite path with at least one edge, with the same rule for apartments.
It contains no bi-infinite path, so clause 2 fails for every pair of edges.

For a field $F$, the projective line over $F$, as a set of points with all two-point subsets as apartments, is a thick building of rank $1$ for $\mathrm{GL}_2(F)$.