From geometry to arithmetic of compact hyperbolic Coxeter polytopes

Nikolay, Bogachev,

doi:10.48550/arxiv.2003.11944

Cited by 1 publication

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References 14 publications

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“…Generally speaking, there are two different approaches to both problems: classification of finite-volume Coxeter polytopes of some certain combinatorial types (see [Kap74, Ess96, Tum07, FT08, FT09, JT18, MZ22, Bur22]) and the theory of arithmetic hyperbolic reflection groups (see [Vin72,Bel16,Bog17,BP18,Bog19,Bog20]). In particular, in the context of arithmetic and quasi-arithmetic reflection groups several authors constructed novel Coxeter polytopes as faces or reflection centralizers of some higher dimensional polytopes (see [Bor87,All06,All13,BK21,BBKS21]).…”

Section: Introductionmentioning

confidence: 99%

Lannér diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes

Alexandrov¹

2022

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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.

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