The Inradius of a Hyperbolic Truncated nn-Simplex

Hyperbolic truncated simplices are polyhedra bounded by at most $$2n+2$$hyperplanes in hyperbolic $$n$$-space. They provide important models in the context of hyperbolic space forms of small volume. In this work, we derive an explicit formula for their inradius by algebraic means and by using the concept of reduced Gram matrix. As an illustration, we discuss implications for some polyhedra related to small volume arithmetic orientable hyperbolic orbifolds

New contributions to hyperbolic polyhedra, reflection groups, and their commensurability

Les groupes de Coxeter hyperboliques forment une classe importante de sous-groupes discrets de Isom (Hn) : ils ont une présentation simple, satisfont des propriétés combinatoires et algébriques agréables, et fournissent des exemples de n-orbifolds hyperboliques de petit volume. Cependant, ils sont loin d'être classifiés, et plusieurs de leurs propriétés restent cryptiques. Ainsi, l'étude des groupes de Coxeter hyperboliques et des polyèdres de Coxeter correspondants est un domaine riche et diversifié, recelant de nombreux problèmes ouverts. Dans ce travail, on résout les trois problèmes suivants : (P1) Trouver une borne dimensionnelle supérieure pour l'existence d'hypercubes de Coxeter hyperboliques, et classifier les hypercubes de Coxeter idéaux. (P2) Trouver le rayon inscrit d'un simplexe tronqué hyperbolique. (P3) Classifier à commensurabilité près les groupes de Coxeter hyperboliques pyramidaux. Nos résultats sont inspirés de travaux précédents respectivement dus à Felikson-Tumarkin [21], Milnor [47], Vinberg [65], Maclachlan [39] et Johnson-Kellerhals-Ratcliffe-Tschantz [31]. Notre solution au problème (P2) a été partiellement publiée dans [29]. De plus, la solution du problème (P3) résulte d'un travail commun avec Rafael Guglielmetti et Ruth Kellerhals [24]

The Inradius of a Hyperbolic Truncated nn n -Simplex

Hyperbolic truncated simplices are polyhedra bounded by at most $$2n+2$$ 2 n + 2 hyperplanes in hyperbolic $$n$$ n -space. They provide important models in the context of hyperbolic space forms of small volume. In this work, we derive an explicit formula for their inradius by algebraic means and by using the concept of reduced Gram matrix. As an illustration, we discuss implications for some polyhedra related to small volume arithmetic orientable hyperbolic orbifolds

Commensurability of hyperbolic Coxeter groups: theory and computation

For hyperbolic Coxeter groups of finite covolume we review and present new theoretical and computational aspects of wide commensurability. We discuss separately the arithmetic and the non-arithmetic cases. Some worked examples are added as well as a panoramic viewto hyperbolic Coxeter groups and their classification

On the Robustness of Democratic Electoral Processes to Computational Propaganda

There is growing evidence of systematic attempts to influence democratic elections by controlled and digitally organized dissemination of fake news. This raises the question of the intrinsic robustness of democratic electoral processes against external influences. Particularly interesting is to identify the social characteristics of a voter population that renders it more resilient against opinion manipulation. Equally important is to determine which of the existing democratic electoral systems is more robust to external influences. Here we construct a mathematical electoral model to address these two questions. We find that electorates are more resilient against opinion manipulations (i) if they are less polarized and (ii) when voters interact more with each other, regardless of their opinion differences, and that (iii) electoral systems based on proportional representation are generally the most robust. Our model qualitatively captures the volatility of the US House of Representatives elections. We take this as a solid validation of our approach.Comment: Main text: 26 pages, 6 figures. Supplementary information: 14 pages, 9 figure

On commensurable hyperbolic Coxeter groups

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space Hn, new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions

Commensurability of hyperbolic Coxeter groups: theory and computation (Geometry and Analysis of Discrete Groups and Hyperbolic Spaces)

"Geometry and Analysis of Discrete Groups and Hyperbolic Spaces". June 22～26, 2015. edited by Michihiko Fujii, Nariya Kawazumi and Ken'ichi Ohshika. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.For hyperbolic Coxeter groups of finite covolume we review and present new theoretical and computational aspects of wide commensurability. We discuss separately the arithmetic and the non-arithmetic cases. Some worked examples are added as well as a panoramic view to hyperbolic Coxeter groups and their classification

Polyèdres et commensurabilité

Le but de cet article est de présenter la notion de commensurabilité de polyèdres. Cette notion est basée uniquement sur l’utilisation de ciseaux et de colle. Les deux premières sections sont élémentaires et servent à présenter un exemple concret et sa formalisation. Dans la dernière section, certains liens avec d’autres domaines des mathématiques sont discutés. Cet article est une traduction abrégée du "Snapshot of Modern Mathematics MFO" [1] des mêmes auteurs