Height Estimates for Bianchi Groups

Author(s)

Doria, Cayo ^[1] ; Paula, Gisele Teixeira ^[2]

Affiliation

^[1] IME USP, Dept Matemat Aplicada, Rua Matao 1010, Cidade Univ, BR-05508090 Sao Paulo, SP - Brazil

^[2] DMAT CCE UFES, Dept Matemat, Av Fernando Ferrari 514, Campus Goiabeiras, BR-29075910 Vitoria, ES - Brazil

• Document type: Journal article
• Source: BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY; v. 52, n. 3, p. 613-627, SEP 2021.
• Web of Science Citations: 0

Abstract

We study the action ofBianchi groups on the hyperbolic 3-spaceH(3). Given the standard fundamental domain for this action and any point in 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 (AU)

• FAPESP's process: 18/15750-9 - Closed curves on hyperbolic manifolds.
• Grantee: Cayo Rodrigo Felizardo Dória
• Support Opportunities: Scholarships in Brazil - Post-Doctoral