1. CJM 2013 (vol 66 pp. 354)
 Kellerhals, Ruth; Kolpakov, Alexander

The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3space
Due to work of W. Parry it is known that the growth
rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$
is a Salem number. This being the arithmetic situation, we prove that the simplex group
(3,5,3) has smallest growth rate among all cocompact hyperbolic
Coxeter groups, and that it is as such unique.
Our approach provides a different proof for
the analog situation in ${\mathbb H^2}$
where E. Hironaka identified Lehmer's number as the minimal growth
rate among all cocompact planar hyperbolic Coxeter groups and showed
that it is (uniquely) achieved by the Coxeter triangle group (3,7).
Keywords:hyperbolic Coxeter group, growth rate, Salem number Categories:20F55, 22E40, 51F15 

2. CJM 2012 (vol 65 pp. 222)
 Sauer, N. W.

Distance Sets of Urysohn Metric Spaces
A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every
isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of
$\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto
$\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal}
if it isometrically embeds every finite metric space $\mathrm{F}$ with
$\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With
$\operatorname{dist}(\mathrm{M})$ being the set of distances between points in
$\mathrm{M}$.)
A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if
it is homogeneous, universal, separable and complete. (It is not
difficult to deduce
that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds
every separable metric space $\mathrm{M}$ with
$\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.)
The main results are: (1) A characterization of the sets
$\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$.
(2) If $R$ is the distance set of an Urysohn metric space and
$\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any
cardinality with distances in $R$, then they amalgamate disjointly to
a metric space with distances in $R$. (3) The completion of every
homogeneous, universal, separable metric space $\mathrm{M}$ is
homogeneous.
Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability Categories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99 

3. CJM 2011 (vol 64 pp. 1359)
 Nozaki, Hiroshi; Sawa, Masanori

Note on Cubature Formulae and Designs Obtained from Group Orbits
In 1960,
Sobolev proved that for a finite reflection group $G$,
a $G$invariant cubature formula is of degree $t$ if and only if
it is exact for all $G$invariant polynomials of degree at most $t$.
In this paper,
we find some observations on invariant cubature formulas and Euclidean designs
in connection with the Sobolev theorem.
First, we give an alternative proof of
theorems by Xu (1998) on necessary and sufficient conditions
for the existence of cubature formulas with some strong symmetry.
The new proof is shorter and simpler compared to the original one by Xu, and
moreover gives a general interpretation of
the analyticallywritten conditions of Xu's theorems.
Second,
we extend a theorem by Neumaier and Seidel (1988) on
Euclidean designs to invariant Euclidean designs, and thereby
classify tight Euclidean designs obtained from
unions of the orbits of the corner vectors.
This result generalizes a theorem of Bajnok (2007) which classifies
tight Euclidean designs invariant under the Weyl group of type $B$
to other finite reflection groups.
Keywords:cubature formula, Euclidean design, radially symmetric integral, reflection group, Sobolev theorem Categories:65D32, 05E99, 51M99 

4. CJM 2011 (vol 64 pp. 123)
 Lee, JaeHyouk

Gosset Polytopes in Picard Groups of del Pezzo Surfaces
In this article, we study the correspondence between the geometry of
del Pezzo surfaces $S_{r}$ and the geometry of the $r$dimensional Gosset
polytopes $(r4)_{21}$. We construct Gosset polytopes $(r4)_{21}$ in
$\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify
divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a1)$simplexes ($a\leq
r$), $(r1)$simplexes and $(r1)$crosspolytopes of the polytope $(r4)_{21}$.
Then we explain how these classes correspond to skew $a$lines($a\leq r$),
exceptional systems, and rulings, respectively.
As an application, we work on the monoidal transform for lines to study the
local geometry of the polytope $(r4)_{21}$. And we show that the Gieser transformation
and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and
$4_{21}$, respectively.
Categories:51M20, 14J26, 22E99 

5. CJM 2011 (vol 63 pp. 1254)
 D'Azevedo, Antonio Breda; Jones, Gareth A.; Schulte, Egon

Constructions of Chiral Polytopes of Small Rank
An abstract polytope of rank $n$ is said to be chiral if its
automorphism group has precisely two orbits on the flags, such that
adjacent flags belong to distinct orbits. This paper describes
a general method for deriving new finite chiral polytopes from old
finite chiral polytopes of the same rank. In particular, the technique
is used to construct many new examples in ranks $3$, $4$, and $5$.
Keywords:abstract regular polytope, chiral polytope, chiral maps Categories:51M20, 52B15, 05C25 

6. CJM 2009 (vol 61 pp. 1300)
 Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c

Monodromy Groups and SelfInvariance
For every polytope $\mathcal{P}$ there is the universal regular
polytope of the same rank as $\mathcal{P}$ corresponding to the
Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given
automorphism $d$ of $\mathcal{C}$, using monodromy groups, we
construct a combinatorial structure $\mathcal{P}^d$. When
$\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that
$\mathcal{P}$ is selfinvariant with respect to $d$, or
$d$invariant. We develop algebraic tools for investigating these
operations on polytopes, and in particular give a criterion on the
existence of a $d$\nobreakdashauto\morphism of a given order. As an application,
we analyze properties of selfdual edgetransitive polyhedra and
polyhedra with two flagorbits. We investigate properties of medials
of such polyhedra. Furthermore, we give an example of a selfdual
equivelar polyhedron which contains no polarity (duality of order
2). We also extend the concept of Petrie dual to higher dimensions,
and we show how it can be dealt with using selfinvariance.
Keywords:maps, abstract polytopes, selfduality, monodromy groups, medials of polyhedra Categories:51M20, 05C25, 05C10, 05C30, 52B70 

7. CJM 2009 (vol 61 pp. 1407)
 Will, Pierre

Traces, CrossRatios and 2Generator Subgroups of $\SU(2,1)$
In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.
Categories:14L24, 22E40, 32M15, 51M10 

8. 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 

9. CJM 2004 (vol 56 pp. 1068)
 Steinbach, Anja; Van Maldeghem, Hendrik

Regular Embeddings of Generalized Hexagons
We classify the generalized hexagons which are laxly
embedded in projective space such that the embedding is flat and
polarized. Besides the standard examples related to the hexagons
defined over the algebraic groups of type $\ssG_2$, $^3\ssD_4$ and
$^6\ssD_4$ (and occurring in projective dimensions $5,6,7$), we
find new examples in unbounded dimension related to the mixed
groups of type $\ssG_2$.
Keywords:Moufang generalized hexagons, embeddings, mixed hexagons, classical, hexagons Categories:51E12, 51A45 

10. CJM 2004 (vol 56 pp. 310)
 Llibre, Jaume; Schlomiuk, Dana

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order
In this article we determine the global geometry of the planar
quadratic differential systems with a weak focus of third order. This
class plays a significant role in the context of Hilbert's 16th
problem. Indeed, all examples of quadratic differential systems with
at least four limit cycles, were obtained by perturbing a system in
this family. We use the algebrogeometric concepts of divisor and
zerocycle to encode global properties of the systems and to give
structure to this class. We give a theorem of topological
classification of such systems in terms of integervalued affine
invariants. According to the possible values taken by them in this
family we obtain a total of $18$ topologically distinct phase
portraits. We show that inside the class of all quadratic systems
with the topology of the coefficients, there exists a neighborhood of
the family of quadratic systems with a weak focus of third order and
which may have graphics but no polycycle in the sense of \cite{DRR}
and no limit cycle, such that any quadratic system in this
neighborhood has at most four limit cycles.
Categories:34C40, 51F14, 14D05, 14D25 

11. CJM 2003 (vol 55 pp. 1080)
 Kellerhals, Ruth

Quaternions and Some Global Properties of Hyperbolic $5$Manifolds
We provide an explicit thick and thin decomposition for oriented
hyperbolic manifolds $M$ of dimension $5$. The result implies improved
universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$
compact, new estimates relating the injectivity radius and the diameter
of $M$ with $\rmvol_5(M)$. The quantification of the thin part is
based upon the identification of the isometry group of the universal
space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of
quaternionic $2\times 2$matrices with Dieudonn\'e determinant
$\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm
L} (2,\mathbb{H})$.
Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40 

12. 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 

13. 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 

14. CJM 1999 (vol 51 pp. 1240)
15. CJM 1999 (vol 51 pp. 1230)
16. CJM 1999 (vol 51 pp. 1175)
 Lehrer, G. I.; Springer, T. A.

Reflection Subquotients of Unitary Reflection Groups
Let $G$ be a finite group generated by (pseudo) reflections in a
complex vector space and let $g$ be any linear transformation which
normalises $G$. In an earlier paper, the authors showed how to
associate with any maximal eigenspace of an element of the coset
$gG$, a subquotient of $G$ which acts as a reflection group on the
eigenspace. In this work, we address the questions of
irreducibility and the coexponents of this subquotient, as well as
centralisers in $G$ of certain elements of the coset. A criterion
is also given in terms of the invariant degrees of $G$ for an
integer to be regular for $G$. A key tool is the investigation of
extensions of invariant vector fields on the eigenspace, which
leads to some results and questions concerning the geometry of
intersections of invariant hypersurfaces.
Categories:51F15, 20H15, 20G40, 20F55, 14C17 

17. CJM 1999 (vol 51 pp. 1149)
 Cohen, A. M.; Cuypers, H.; Sterk, H.

Linear Groups Generated by Reflection Tori
A reflection is an invertible linear transformation of a vector
space fixing a given hyperplane, its axis, vectorwise and a given
complement to this hyperplane, its center, setwise. A reflection
torus is a onedimensional group generated by all reflections with
fixed axis and center.
In this paper we classify subgroups of general linear groups (in
arbitrary dimension and defined over arbitrary fields) generated by
reflection tori.
Categories:20Hxx, 20Gxx, 51A50 

18. CJM 1999 (vol 51 pp. 1277)
 Shephard, G. C.

Isomorphism Invariants for Projective Configurations
An isomorphism invariant is an expression, defined for a
configuration in the projective plane, which takes the same value
for all isomorphic configurations. Examples are given as well as a
general method (Nehring sequences) for constructing such
invariants.
Category:51N15 

19. 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 

20. CJM 1998 (vol 50 pp. 426)
21. CJM 1997 (vol 49 pp. 1162)
 Ku, HsuTung; Ku, MeiChin; Zhang, XinMin

Isoperimetric inequalities on surfaces of constant curvature
In this paper we introduce the concepts of hyperbolic and elliptic
areas and prove uncountably many new geometric isoperimetric
inequalities on the surfaces of constant curvature.
Keywords:Gaussian curvature, GaussBonnet theorem, polygon, pseudopolygon, pseudoperimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Categories:51M10, 51M25, 52A40, 53C20 
