On the trace fields of hyperbolic Dehn fillings

TL;DR

Under the assumption of Lehmer's conjecture, the paper provides bounds on the degree of the trace field of hyperbolic Dehn fillings of 3-manifolds, relating it to the filling parameters and a constant depending on the manifold.

Contribution

It establishes a conditional estimate for the degree of trace fields of hyperbolic Dehn fillings based on the parameters and Lehmer's conjecture, linking number theory and 3-manifold geometry.

Findings

Bounds on trace field degree in terms of filling parameters

Dependence of bounds on a manifold-specific constant

Conditional estimates assuming Lehmer's conjecture

Abstract

Assuming Lehmer's conjecture, we estimate the degree of the trace field $K(M_{p/q})$ of a hyperbolic Dehn-filling $M_{p/q}$ of a 1-cusped hyperbolic 3-manifold $M$ by $$ \dfrac{1}{C}(\max\;\{|p|,|q|\})\leq \text{deg }K(M_{p/q}) \leq C(\max\;\{|p|,|q|\}) $$ where $C=C_M$ is a constant that depends on $M$.