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Hyperbolic tessellations and generators of 
${K}_{\textbf {3}}$ for imaginary quadratic fields
Part of:
$K$-theory in number theory
Real and complex geometry
Algebraic number theory: global fields
Higher algebraic $K$-theory
Other groups of matrices
Arithmetic algebraic geometry
Published online by Cambridge University Press: 24 May 2021
Abstract
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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.
- Type
- Algebra
- Information
- Creative Commons
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