A Volume Function for CR Tetrahedra

TL;DR

This paper introduces a volume function for CR tetrahedra in the 3-sphere, invariant under automorphisms, satisfying a five-term relation that links CR and hyperbolic geometries.

Contribution

It defines a new volume function for CR tetrahedra invariant under PU(2,1) and establishes its five-term relation connecting CR and hyperbolic geometries.

Findings

The volume function is invariant under PU(2,1) transformations.

The volume satisfies a five-term relation.

The construction involves the Bloch-Wigner dilogarithm.

Abstract

We define a volume function on configurations of four points in the 3-sphere which is invariant under the action of PU(2,1), the automorphism group of the CR structure defined on the sphere by its embedding in complex 2-space. We show that the volume function, constructed using appropriate combinations of the dilogathm function of Bloch-Wigner, satisfies a five term relation in a more general context which includes at the same time CR and real hyperbolic geometry.