Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields

TL;DR

This paper introduces new methods using hyperbolic tessellations and calculations in pre-Bloch groups to explicitly construct generators of K_3-groups for imaginary quadratic fields, providing the first proven examples of such generators.

Contribution

It develops novel techniques for constructing explicit K_3-group generators, improves Bloch group theory, and confirms predictions related to the Lichtenbaum conjecture for abelian number fields.

Findings

First explicit generators of K_3 for imaginary quadratic fields

Validated the Lichtenbaum conjecture prediction for abelian fields

Enhanced understanding of Bloch groups and their applications

Abstract

We develop methods for constructing explicit generators, modulo torsion, of the K_3-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3-space or on direct calculations in suitable pre-Bloch groups, and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite K_3-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K_3 of any field, predict the precise power of 2 that should occur in the Lichtenbaum conjecture at -1 and prove that the latter prediction is valid for all abelian number fields.