1. CJM 2009 (vol 61 pp. 740)
 Caprace, PierreEmmanuel; Haglund, Frédéric

On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
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 Gromovhyperbolic 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.
Keywords:Coxeter group, flat rank, $\cat0$ space, building Categories:20F55, 51F15, 53C23, 20E42, 51E24 

2. CJM 2002 (vol 54 pp. 239)
 Cartwright, Donald I.; Steger, Tim

Elementary Symmetric Polynomials in Numbers of Modulus $1$
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 ${\tilde A}_n$.
Categories:05E05, 33C45, 30C15, 51E24 

3. CJM 2001 (vol 53 pp. 809)
 Robertson, Guyan; Steger, Tim

Asymptotic $K$Theory for Groups Acting on $\tA_2$ Buildings
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 BruhatTits 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 CuntzKrieger algebras.
Keywords:$K$theory, $C^*$algebra, affine building Categories:46L80, 51E24 

4. CJM 1999 (vol 51 pp. 347)
 Mühlherr, Bernhard; Van Maldeghem, Hendrik

Exceptional Moufang Quadrangles of Type $\mathsf{F}_4$
In this paper, we present a geometric construction of the Moufang
quadrangles discovered by Richard Weiss (see Tits \& Weiss
\cite{TitWei: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.
Categories:51E12, 51E24 
