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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © Sociedade Brasileira de Matemática 2020
- ISSN
- 1678-7544
- eISSN
- 1678-7714
- DOI
- 10.1007/s00574-020-00222-9
- Publisher site
- See Article on Publisher Site
Abstract
We study the action of Bianchi groups on the hyperbolic 3-space H3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^3 $$\end{document}. Given the standard fundamental domain for this action and any point in H3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathbb {H}^3 $$\end{document}, we show that there exists an element in the group which sends the given point into the fundamental domain such that its height is bounded by a quadratic function on the coordinates of the point. This generalizes and establishes a sharp version of a similar result of Habegger and Pila for the action of the Modular group on the hyperbolic plane. Our main theorem can be applied in the reduction theory of binary Hermitian forms with entries in the ring of integers of quadratic imaginary fields. We also show that the asymptotic behavior of the number of elements in a fixed Bianchi group with height at most T is biquadratic in T.
Journal
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Jul 23, 2020
Keywords: Bianchi groups; Heights; Hermitian forms; 11E57; 11H55