Hyperbolic three manifolds of bounded volume and trace field degree II

TL;DR

This paper proves the Bounded Height Conjecture, establishing that only finitely many hyperbolic three-manifolds have both bounded volume and trace field degree, advancing understanding in hyperbolic geometry and number theory.

Contribution

The paper proves the Bounded Height Conjecture, linking geometric and arithmetic properties of hyperbolic 3-manifolds for the first time.

Findings

Finiteness of hyperbolic 3-manifolds with bounded volume and trace field degree

Validation of the Bounded Height Conjecture in hyperbolic geometry

Implications for classification of hyperbolic 3-manifolds

Abstract

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.