Abstract
We study the action of Bianchi groups on the hyperbolic 3-space \(\mathbb {H}^3 \). Given the standard fundamental domain for this action and any point in \( \mathbb {H}^3 \), 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.
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Acknowledgements
We would like to thank Mikhail Belolipetsky for his encouragement and interest in this work. We also thank the referee for numerous comments and suggestions, including pointing out gaps and possible improvements on our main result. We are grateful to Universidade de São Paulo and Universidade Federal do Espírito Santo for the support and hospitality. Besides, the second author acknowledges the University of Lille for receiving her in her postdoctoral stage and for the good interchange of ideas that she had there.
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C. Dória is grateful for the support of FAPESP Grant 2018/15750-9. G. T. Paula was partially supported by Math-AmSud Project 88887.199703/2018-00 from CAPES.
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Dória, C., Paula, G.T. Height Estimates for Bianchi Groups. Bull Braz Math Soc, New Series 52, 613–627 (2021). https://doi.org/10.1007/s00574-020-00222-9
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DOI: https://doi.org/10.1007/s00574-020-00222-9