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

Part of: $K$-theory in number theory Higher algebraic $K$-theory Algebraic number theory: global fields Arithmetic algebraic geometry Real and complex geometry Other groups of matrices

Published online by Cambridge University Press:  24 May 2021

David Burns Open the ORCID record for David Burns [Opens in a new window] ,
Rob de Jeu Open the ORCID record for Rob de Jeu [Opens in a new window] ,
Herbert Gangl Open the ORCID record for Herbert Gangl [Opens in a new window] ,
Alexander D. Rahm Open the ORCID record for Alexander D. Rahm [Opens in a new window]  and
Dan Yasaki Open the ORCID record for Dan Yasaki [Opens in a new window]
David Burns
Affiliation:
King’s College London, Dept. of Mathematics, LondonWC2R 2LS, United Kingdom; E-mail: david.burns@kcl.ac.uk
Rob de Jeu
Affiliation:
Faculteit der Bètawetenschappen, Afdeling Wiskunde, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081HV, Amsterdam, The Netherlands; E-mail: r.m.h.de.jeu@vu.nl
Herbert Gangl
Affiliation:
Department of Mathematical Sciences, Stockton Road, DurhamDH1 3LE, United Kingdom; E-mail: herbert.gangl@durham.ac.uk
Alexander D. Rahm
Affiliation:
Laboratoire de mathématiques GAATI, Université de la Polynésie française, BP6570, 98702Faaa, French Polynesia; E-mail: rahm@gaati.org
Dan Yasaki
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Greensboro, PO Box 26170, Greensboro, NC27402-6170, USA; E-mail: d_yasaki@uncg.edu

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.

MSC classification

Primary: 11G55: Polylogarithms and relations with $K$-theory 11R70: $K$-theory of global fields 19F27: Étale cohomology, higher regulators, zeta and $L$-functions 19D45: Higher symbols, Milnor $K$-theory
Secondary: 11R42: Zeta functions and $L$-functions of number fields 20H20: Other matrix groups over fields 51M20: Polyhedra and polytopes; regular figures, division of spaces
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e40
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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