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Hyperbolic tessellations and generators of for imaginary quadratic fields

Burns, David and de Jeu, Rob and Gangl, Herbert and Rahm, Alexander D. and Yasaki, Dan (2021) 'Hyperbolic tessellations and generators of for imaginary quadratic fields.', Forum of Mathematics, Sigma, 9 . e40.


We develop methods for constructing explicit generators, modulo torsion, of the K3 -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 K3 -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K3 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.

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Publisher statement:This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (, which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Date accepted:No date available
Date deposited:01 November 2021
Date of first online publication:24 May 2021
Date first made open access:01 November 2021

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