---
tag: 6BQJ50NEJ0
kind: concept
name: Building
msc: ["20E42", "51E24"]
sources:
  - url: https://en.wikipedia.org/wiki/Building_(mathematics)
same_as:
  wikidata: Q4986445
  wikipedia: Building (mathematics)
---

## Summary

A building is a [[simplicial complex|V8GZ0W1ZZ5]] glued together from copies of a Coxeter complex, called apartments, under strong compatibility rules.
Jacques Tits introduced buildings to give a geometric picture of reductive algebraic groups and of the [[groups of Lie type|NPEBABQ49M]].
Each group with a BN-pair acts on a building, and the [[Bruhat decomposition|T91E86D0RH]] becomes a statement about the relative position of two chambers.
Buildings of $p$-adic groups are spaces of nonpositive curvature, used much like symmetric spaces for real Lie groups.

## Explanations

### L1 human

Think of a building as a large structure made of identical flat floors, the apartments.
Any two rooms of the structure lie together on at least one floor.
Two floors that share two rooms can be laid on top of each other without moving those rooms.
The simplest infinite example is a [[tree|062D9KXJB1]] in which every vertex has at least three neighbours.
Its apartments are the infinite straight lines through the tree.