Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if $\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0) realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where $z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which $z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${\tilde A}_n$.

Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras.

In this paper, we present a geometric construction of the Moufang quadrangles discovered by Richard Weiss (see Tits \& Weiss \cite{Tit-Wei:97} or Van Maldeghem \cite{Mal:97}). The construction uses fixed point free involutions in certain mixed quadrangles, which are then extended to involutions of certain buildings of type $\ssF_4$. The fixed flags of each such involution constitute a generalized quadrangle. This way, not only the new exceptional quadrangles can be constructed, but also some special type of mixed quadrangles.