Introduction

Given a finitely generated Coxeter group W, I described, in [D, Section 14], a certain contractible simplicial complex, here denoted |W|, on which W acts properly with compact quotient. After writing [D], I realized that there was a similarly defined, contractible simplicial complex associated to any building C. This complex is here denoted |C| and called the “geometric realization” of C. The definition is such that the geometric realization of each apartment is isomorphic to |W|. (N. B. Our terminology does not agree with standard usage. For example, if W is finite, then the usual Coxeter complex of W is homeomorphic to a sphere, while our |W| is homeomorphic to the cone on this sphere.)

There is a natural piecewise Euclidean metric on |W| (described in §9) so that W acts as a group of isometries. Following an idea of Gromov ([G, pp. 131–132]), Gabor Moussong proved in his Ph.D. thesis [M] that with this metric |W| is “CAT(0)” (in the sense of [G]). This is equivalent to saying that it is simply connected and “nonpositively curved”. Moussong's result implies, via a standard argument, the following theorem.

Theorem. The (correctly defined) geometric realization of any building is CAT(0).

Although this theorem was known to Moussong, it is not included in [M].

The theorem implies, for example, that the Bruhat Tits Fixed Point Theorem can be applied to any building. (See Corollary 11.9.)

One of the purposes of this paper is to provide the “correct definition” of the geometric realization |C| and to give the “standard argument” for deducing the above theorem from Moussong's result.