The minimal covolume right-angled Coxeter group in hyperbolic 3-space

2025/09/11 by Egorov, A., Vesnin, A.
#51F15 #57K32 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2509.09261

Abstract

We prove that among all right-angled Coxeter groups in hyperbolic 3-space, the group generated by reflections in the faces of a right-angled triangular bipyramid with three ideal and two finite vertices has the smallest covolume. The group is arithmetic and its covolume equals Catalan's constant G = 0.915965....

Citations

• Geometric and arithmetic properties of Löbell polyhedra
• On volumes of hyperbolic right-angled polyhedra
• The non-arithmetic cusped hyperbolic 3-orbifold of minimal volume
• Volume estimates for right-angled hyperbolic polyhedra
• Hyperbolic orbifolds of small volume
• The number of cusps of right-angled polyhedra in hyperbolic spaces
• Computing arithmetic invariants for hyperbolic reflection groups
• Minimum volume cusped hyperbolic three-manifolds
• Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds
• Andreev's Theorem on hyperbolic polyhedra

Related

• On the commensurability of hyperbolic Coxeter groups
• Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups
• On volumes of hyperbolic Coxeter polytopes and quadratic forms
• Relative hyperbolicity and right-angled Coxeter groups
• The non-arithmetic cusped hyperbolic 3-orbifold of minimal volume