Buildings are \(\text{CAT}(0)\). (English) Zbl 0978.51005

Davis, Michael W.

Kropholler, Peter H. (ed.) et al., Geometry and cohomology in group theory. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 252, 108-123 (1998).

The title of the paper sounds paradoxical, since a well-known result of Solomon and Tits says that the geometric realization of the flag complex of a spherical building is homotopy equivalent to a wedge of spheres. The point is that Davis constructs a somewhat different simplicial complex from a building.
Let \({\mathcal C}\) be a building (of finite rank), viewed as a chamber system over a set \(I\) as in [M. Ronan, Lectures on buildings, Academic Press (1989; Zbl 0694.51001)]. Let \(S\) denote the poset of all spherical residues of \({\mathcal C}\) (including \({\mathcal C}\) if \({\mathcal C}\) is spherical), ordered by inclusion. The derived poset \(S'\) of \(S\) (i.e. the collection of all chains in \(S)\) is an abstract simplicial complex. Davis proves that the geometric realization \(|S'|\) of \(S'\) is a CAT(0) space; in particular, \(|S'|\) is contractible.
This construction should be compared to the simplicial complex traditionally associated to a building: here, \(R\) is the poset of all proper residues of \({\mathcal C}\) (excluding \({\mathcal C})\), ordered by reversed inclusion, see [M. Ronan, loc. cit.]. This poset, which is an abstract simplicial complex, is the flag complex of \({\mathcal C}\). Note that the derived poset \(R'\) (which is more like Davis’ \(S')\) is just the barycentric subdivision of \(R\) and so \(|R|\) is homeomorphic to \(|R'|\). If \({\mathcal C}\) is affine and irreducible, then \(R'=S'\). On the other hand, if \({\mathcal C}\) is spherical, then \(|S'|\) is the cone over \(|R'|\) (the tip of the cone is \(\{{\mathcal C}\})\).
As an application in geometric group theory, Davis shows that certain spaces arising from graph products of groups which were studied by J. Harlander and H. Meinert [J. Lond. Math. Soc., II. Ser. 53, No. 1, 99-117 (1996; Zbl 0854.20062)] are geometric realizations (in Davis’ sense) of right-angled buildings.
For the entire collection see [Zbl 0991.00031].

Reviewer: Linus K.H.Kramer (Würzburg)

MSC:

51E24	Buildings and the geometry of diagrams
20F65	Geometric group theory
20E42	Groups with a \(BN\)-pair; buildings
57M07	Topological methods in group theory

Keywords:

flag complex; spherical building; simplicial complex

Citations:

Zbl 0694.51001; Zbl 0854.20062