Group theory and generalizations Definition

Building

Builds on Simplicial complex Coxeter group Tree

Definition

Let $(W, S)$ be a Coxeter system. 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 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:

Each apartment is isomorphic to a Coxeter complex.
For any two simplices $A$ and $B$ of $\Delta$ , there is an apartment that contains both.
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 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 $\mathrm{GL}_2(F)$ .

Reading path

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Coxeter group concept
Simplicial complex concept
Tree concept
Building definition

Tag	`70Q4YVF322`
URI	`m2://70Q4YVF322`
Kind	definition
Writing	W0 stub
MSC	20E42, 51E24
Defines	Building
Uses	Simplicial complex, Coxeter group, Tree
Changed	2 November 2026

Building

Definition

Remarks

Reading path

Further reading

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