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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Integral binary Hamiltonian forms and their waterworlds
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by Jouni Parkkonen and Frédéric Paulin PDF
Conform. Geom. Dyn. 25 (2021), 126-169 Request permission


We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathscr {O}$ in a definite quaternion algebra over $\mathbb {Q}$, we define the waterworld of $f$, analogous to Conway’s river and Bestvina-Savin’s ocean, and use it to give a combinatorial description of the values of $f$ on $\mathscr {O}\times \mathscr {O}$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the $\operatorname {SL}_{2}(\mathscr {O})$-equivariant Ford-Voronoi cellulation of the real hyperbolic $5$-space, and the conformal action of $\operatorname {SL}_{2}(\mathscr {O})$ on the Hamilton quaternions.
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Additional Information
  • Jouni Parkkonen
  • Affiliation: Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland
  • MR Author ID: 603219
  • ORCID: 0000-0003-2132-0567
  • Email:
  • Frédéric Paulin
  • Affiliation: Laboratoire de mathématique d’Orsay, UMR 8628 CNRS, Université Paris-Saclay, Bâtiment 307, 91405 ORSAY Cedex, France
  • ORCID: 0000-0002-2270-5744
  • Email:
  • Received by editor(s): September 27, 2020
  • Received by editor(s) in revised form: July 4, 2021
  • Published electronically: October 20, 2021
  • Additional Notes: This work was supported by the French-Finnish CNRS grant PICS No. 6950. The second author greatly acknowledges the financial support of Warwick University for a one month stay, decisive for the writing of this paper
  • © Copyright 2021 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 25 (2021), 126-169
  • MSC (2020): Primary 11E39, 20G20, 11R52, 53A35, 15A21, 11F06, 20H10
  • DOI:
  • MathSciNet review: 4328674