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  "body": "\n## Definition\n\nLet $(W, S)$ be a [[Coxeter system|G6FND7X69J]].\nFor $J \\subseteq S$, let $W_J$ be the subgroup of $W$ generated by $J$.\nThe Coxeter complex $\\Sigma(W, S)$ is the [[simplicial complex|V8GZ0W1ZZ5]] whose simplices are the cosets $w W_J$ with $w \\in W$ and $J \\subseteq S$.\nA simplex $w W_J$ is a face of $w' W_{J'}$ when $w W_J \\supseteq w' W_{J'}$.\nA *building* is a simplicial complex $\\Delta$ together with a family of subcomplexes, called *apartments*, such that:\n\n1. Each apartment is isomorphic to a Coxeter complex.\n2. For any two simplices $A$ and $B$ of $\\Delta$, there is an apartment that contains both.\n3. 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 $B$.\n\nThe maximal simplices are the *chambers*.\nThe building is *thick* if each face of codimension one of a chamber lies in at least three chambers.\n\n## Remarks\n\nExample: 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 $\\Delta$.\nEach such path is the Coxeter complex of the infinite dihedral group, a line divided into edges.\nClause 2 holds because the path between two edges extends at both ends to a bi-infinite path, since no vertex has degree $1$.\nClause 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.\nEach vertex lies in at least three edges, so the building is thick.\n\nNon-example: a finite path with at least one edge, with the same rule for apartments.\nIt contains no bi-infinite path, so clause 2 fails for every pair of edges.\n\nFor 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 $\\mathrm{GL}_2(F)$.\n",
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