Coxeter groups, the Davis complex, and isolated flats.

Coxter groups arose as a natural generalization of reflection groups. J. Tits defined them in a simple way using generators and relations, that is, using a group presentation \(W \cong ⟨S | R⟩\). Coxeter groups have a wide range of applications; for example, every Weyl group may be realized as a finite, irreducible Coxeter group.

The Davis complex \(\Sigma\) is a geometric realization of Coxeter groups, which is CAT(0) for every Coxeter group. It has therefore been one of the first classes of examples for CAT(0) spaces. We first provide a general introduction to Coxeter groups and the Davis complex, and continue discussing when the Davis complex has so called flats.

Flats are convex subsets which are isometric to \(\mathbb{R}^n\). We say that \(\Sigma\) has isolated flats if there exists a collection \(F\) of flats in \(\Sigma\) , satisfying the isolated property:

1. (A)  There is a constant \( D<\infty\) such that each flat \(F\) of \(\Sigma\) lies in a tubular \(D\)-neighborhood of some \(C \in F\).

2. (B)  For each positive \(r < \infty\), there is a constant \(\rho = \rho(r) < \infty\) so that for any two distincit elements \(C,C′ \in F \) we have \(diam(N_r(C)\cap N_r(C′)) < \rho\), where \(N_r(C)\) denotes the tubular \(r\)-neighborhood of \(C\).

Given a Coxeter group and a set of generators \(S\), we can read from the Coxeter diagram if the resulting Davis complex has isolated flats. This classification is due to work by P. Caprace. We introduce the necessary concepts and give examples of Coxeter groups where \(\Sigma \) has isolated flats.