Height estimates for Bianchi groups

TL;DR

This paper investigates the action of Bianchi groups on hyperbolic 3-space, providing bounds on the height of group elements that map points into a fundamental domain, with applications to reduction theory and counting problems.

Contribution

It establishes a quadratic height bound for Bianchi group actions and analyzes the asymptotic growth of elements with bounded height, extending previous results to hyperbolic 3-space.

Findings

Existence of a height-bounding group element for any point in hyperbolic space.

A sharp quadratic bound generalizing Habegger and Pila's result.

Biquadratic asymptotic growth of elements with height at most T.

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 $T$.