Tetrahedra of flags, volume and homology of SL(3)

TL;DR

This paper introduces a generalized volume concept for flag tetrahedra complexes, unifying hyperbolic and CR tetrahedra volumes, and explores its algebraic properties related to the Bloch group, building on and extending prior foundational work.

Contribution

It defines a new volume for flag tetrahedra complexes, unifies existing volume theories, and relates these to the Bloch group, extending previous results by Neumann-Zagier, Neumann, and Kabaya.

Findings

The volume belongs to the Bloch group under certain conditions.

The approach generalizes classical hyperbolic volume results.

Connections to Fock-Goncharov's work are established.

Abstract

In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to the work of Fock and Goncharov.