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Hyperbolic tessellations and generators of ${K}_{\textbf {3}}$ for imaginary quadratic fields

Published online by Cambridge University Press:  24 May 2021

David Burns
King’s College London, Dept. of Mathematics, LondonWC2R 2LS, United Kingdom; E-mail:
Rob de Jeu
Faculteit der Bètawetenschappen, Afdeling Wiskunde, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081HV, Amsterdam, The Netherlands; E-mail:
Herbert Gangl
Department of Mathematical Sciences, Stockton Road, DurhamDH1 3LE, United Kingdom; E-mail:
Alexander D. Rahm
Laboratoire de mathématiques GAATI, Université de la Polynésie française, BP6570, 98702Faaa, French Polynesia; E-mail:
Dan Yasaki
Department of Mathematics and Statistics, University of North Carolina at Greensboro, PO Box 26170, Greensboro, NC27402-6170, USA; E-mail:


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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.

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