In this paper we study × 0 -products of Lannér diagrams. We prove that every × 0 -product of at least four Lannér diagrams with at least one diagram of order ≥ 3 is superhyperbolic. This fact is used to classify compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of some number of simplices. Such polytopes were already classified in different combinatorial cases: simplices [Lan50], cubes [Poi82, And70a, JT18], simplicial prisms [Kap74], products of two simplices [Ess96], and products of three simplices [Tum07]. As a corollary of our first result mentioned above, we obtain that these classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices.We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order 2. The second result of this paper slightly improves the recent Burcroff upper bound on dimension for such polytopes to 12.
Получены новые верхние оценки объемов прямоугольных многогранников в пространстве Лобачевского $\mathbb{H}^3$ в следующих трех случаях: для идеальных многогранников, все вершины которых лежат на абсолюте, для компактных многогранников, все вершины которых конечны, и для многогранников конечного объема с вершинами обоих типов.
Библиография: 23 названия.
In this paper we study
×
0
\times _0
-products of Lannér diagrams. We prove that every
×
0
\times _0
-product of at least four Lannér diagrams with at least one diagram of order
≥
3
\geq 3
is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices.
We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order
2
2
. The second result of this paper slightly improves recent Burcroff’s upper bound on the dimension of such polytopes to
12
12
.
In this paper we improve known upper bounds on volumes of right-angled polyhedra in hyperbolic space H 3 in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.
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