On hyperbolic Coxeter -cubes

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• 作者：Matthieu Jacquemet ^{jacquemet.matthieu@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：59
• 期：Complete
• 页码：192-203
• 全文大小：470 K

文摘

Besides simplices, n$n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n$n$-cubes are not classified. In this work, we first show that there are no Coxeter n$n$-cubes in Hⁿ${\mathbb{H}}^n$ for n≥10$n\ge 10$. Then, we show that the ideal   ones exist only for n=2$n=2$ and 3$3$, and provide a classification. The methods used are of combinatorial and algebraic nature, using properties of a Coxeter graph, its Schläfli matrix, and the Gram matrix of a polyhedron.