The maximum volume of hyperbolic polyhedra

Belletti, Giulio

doi:10.1090/tran/8215

The maximum volume of hyperbolic polyhedra

Author: Giulio Belletti
Journal: Trans. Amer. Math. Soc. 374 (2021), 1125-1153
MSC (2010): Primary 52B10
DOI: https://doi.org/10.1090/tran/8215
Published electronically: November 3, 2020
MathSciNet review: 4196389
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Abstract:

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal, or hyperideal). We find that the supremum is always equal to the volume of the rectification of the $1$-skeleton.

The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lower-dimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron.

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