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 this prediction is valid for all abelian number fields.
Circulant digraphs integral over number fields
