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<p>It is proved that among all right-angled Coxeter groups in three-dimensional hyperbolic space, the group generated by reflections in the faces of a right-angled triangular bipyramid has the smallest covolume.This bipyramid has three ideal and two finite vertices.The group is arithmetic, and its covolume is equal to Catalan’s constant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ G=0.915965\dots $</EquationSource> </InlineEquation>.</p>

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The Right-Angled Coxeter Group of Minimal Covolume in Three-Dimensional Hyperbolic Space

  • 期刊论文

  • 发表日期：

    2025年11月24日

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  • A. Yu. Vesnin,
  • A. A. Egorov

摘要

It is proved that among all right-angled Coxeter groups in three-dimensional hyperbolic space, the group generated by reflections in the faces of a right-angled triangular bipyramid has the smallest covolume.This bipyramid has three ideal and two finite vertices.The group is arithmetic, and its covolume is equal to Catalan’s constant $ G=0.915965\dots $ .