Seidel's conjectures in hyperbolic 3-space

TL;DR

This paper proves two conjectures by Seidel regarding the volume of ideal hyperbolic tetrahedra, showing it depends monotonically on algebraic properties of the Gram matrix of vertices in hyperbolic 3-space.

Contribution

It establishes the relationship between the volume of ideal hyperbolic tetrahedra and algebraic invariants of their Gram matrices, confirming Seidel's conjectures.

Findings

Volume determined by Gram matrix invariants

Volume is a monotonic function of matrix determinants and permanents

Proof of Seidel's conjectures in hyperbolic 3-space

Abstract

We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in "On the volume of a hyperbolic simplex", Stud. Sci. Math. Hung. 21, 243-249, 1986. These conjectures concern expressing the volume of an ideal hyperbolic tetrahedron as a monotonic function of algebraic maps. More precisely, Seidel's first conjecture states that the volume of an ideal tetrahedron in hyperbolic 3-space is determined by (the permanent and the determinant of) the doubly stochastic Gram matrix $G$ of its vertices; Seidel's fourth conjecture claims that the mentioned volume is a monotonic function of both the permanent and the determinant of $G$.