Hyperbolic three manifolds of bounded volume and trace field degree

TL;DR

This paper explores the finiteness of hyperbolic 3-manifolds with bounded trace field degree, extending Hodgson's results to multi-cusped cases and employing arithmetic geometry techniques.

Contribution

It conjectures a generalization of Hodgson's finiteness result for manifolds with multiple cusps and provides initial positive evidence for the 2-cusped case.

Findings

Finiteness conjecture for multi-cusped hyperbolic 3-manifolds.

Proof for 2-cusped manifolds with linearly independent cusp shapes.

Application of the Bounded Height Conjecture in this context.

Abstract

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a manifold has linearly independent cusp shapes, we show that the manifold has the desired property.To prove the results, we use the proof of the Bounded Height Conjecture in arithmetic geometry.