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1. CMB Online first

Beardon, Alan F.
Non-discrete frieze groups
The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practical way. In addition, this discussion sheds light on why there are only seven conjugacy classes in the classical case.

Keywords:frieze groups, isometry groups
Categories:51M04, 51N30, 20E45

2. CMB 2014 (vol 57 pp. 390)

Morita, Jun; Rémy, Bertrand
Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations
We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).

Keywords:Kac-Moody group, twin tree, simplicity, root system, building
Categories:20G44, 20E42, 51E24

3. CMB 2011 (vol 55 pp. 487)

Deng, Xinghua; Moody, Robert V.
Weighted Model Sets and their Higher Point-Correlations
Examples of distinct weighted model sets with equal $2,3,4, 5$-point correlations are given.

Keywords:model sets, correlations, diffraction
Categories:52C23, 51P05, 74E15, 60G55

4. CMB 2011 (vol 55 pp. 329)

Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.
Non-Discrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,n,\infty;k)$.

Keywords:complex hyperbolic triangle group
Categories:51M10, 32M15, 53C55, 53C35

5. CMB 2010 (vol 53 pp. 534)

Pambuccian, Victor
Acute Triangulation of a Triangle in a General Setting
We prove that, in ordered plane geometries endowed with a very weak notion of orthogonality, one can always triangulate any triangle into seven acute triangles, and, in case the given triangle is not acute, into no fewer than seven.

Categories:51G05, 51F20, 51F05

6. CMB 2010 (vol 53 pp. 394)

Averkov, Gennadiy
On Nearly Equilateral Simplices and Nearly l∞ Spaces
By $\textrm{d}(X,Y)$ we denote the (multiplicative) Banach--Mazur distance between two normed spaces $X$ and $Y.$ Let $X$ be an $n$-dimensional normed space with $\textrm{d}(X,\ell_\infty^n) \le 2,$ where $\ell_\infty^n$ stands for $\mathbb{R}^n$ endowed with the norm $\|(x_1,\dots,x_n)\|_\infty := \max \{|x_1|,\dots, |x_n| \}.$ Then every metric space $(S,\rho)$ of cardinality $n+1$ with norm $\rho$ satisfying the condition $\max D / \min D \le 2/ \textrm{d}(X,\ell_\infty^n)$ for $D:=\{ \rho(a,b) : a, b \in S, \ a \ne b\}$ can be isometrically embedded into $X.$

Categories:52A21, 51F99, 52C99

7. CMB 2009 (vol 52 pp. 435)

Monson, B.; Schulte, Egon
Modular Reduction in Abstract Polytopes
The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind:\ first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}[\tau]$ (with $\tau$ the golden ratio), to construct new regular $4$-polytopes of hyperbolic types $\{3,5,3\}$ and $\{5,3,5\}$ with automorphism groups given by finite orthogonal groups.

Keywords:abstract polytopes, regular and chiral, Coxeter groups, modular reduction
Categories:51M20, 20F55

8. CMB 2009 (vol 52 pp. 407)

Lángi, Zsolt; Naszódi, Márton
On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies
The Bezdek--Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $\Re^d$ is $2^d$. Nasz\'odi proved that the quantity in question is not larger than $2^{d+1}$. We present an improvement to this result by proving the upper bound $3\cdot2^{d-1}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

Keywords:Bezdek--Pach Conjecture, homothets, packing, Hadwiger number, antipodality
Categories:52C17, 51N20, 51K05, 52A21, 52A37

9. CMB 2007 (vol 50 pp. 474)

Zhou, Jiazu
On Willmore's Inequality for Submanifolds
Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$.

Keywords:submanifold, mean curvature, kinematic formul, scalar curvature
Categories:52A22, 53C65, 51C16

10. CMB 1997 (vol 40 pp. 158)

Coxeter, H. S. M.
The trigonometry of hyperbolic tessellations
For positive integers $p$ and $q$ with $(p-2)(q-2) > 4$ there is, in the hyperbolic plane, a group $[p,q]$ generated by reflections in the three sides of a triangle $ABC$ with angles $\pi /p$, $\pi/q$, $\pi/2$. Hyperbolic trigonometry shows that the side $AC$ has length $\psi$, where $\cosh \psi = c/s$, $c = \cos \pi/q$, $s = \sin\pi/p$. For a conformal drawing inside the unit circle with centre $A$, we may take the sides $AB$ and $AC$ to run straight along radii while $BC$ appears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius $1/\sinh \psi = s/z$, where $z = \sqrt{c^2-s^2}$, while its centre is at distance $1/\tanh \psi = c/z$ from $A$. In the hyperbolic triangle $ABC$, the altitude from $AB$ to the right-angled vertex $C$ is $\zeta$, where $\sinh\zeta = z$.

Categories:51F15, 51N30, 52A55

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