The extended Bloch group and algebraic K-theory

TL;DR

This paper introduces an extended Bloch group for any field, providing an explicit algebraic description of K_3^ind(F) for number fields, along with formulas for regulators and applications to hyperbolic 3-manifolds.

Contribution

It defines an extended Bloch group for arbitrary fields and proves its isomorphism to K_3^ind(F) for number fields, offering explicit generators, relations, and regulator formulas.

Findings

Explicit description of K_3^ind(F) for number fields

Concrete regulator formulas and symbol expressions

Application to hyperbolic 3-manifolds with finite volume

Abstract

We define an extended Bloch group for an arbitrary field F, and show that this group is canonically isomorphic to K_3^ind(F) if F is a number field. This gives an explicit description of K_3^ind(F) in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class in K_3^ind(k) tensor Z[1/2].