On the trace fields of hyperbolic Dehn fillings

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 $M$.

Figures (4)

• Figure 1: For instance, if the Newton polygon of the $A$-polynomial $M$ appears as above, then $S_A$ is the slope of the depicted edge.
• Figure 3: In the proof of Theorem \ref{['21012507']}, to verify the claim for $(p,q)$ contained in the shaded region above, we need the estimations carried out in Lemmas \ref{['21013003']}-\ref{['21011701']}.
• Figure 4: We have established the claim for $(p, q)$ in the region bounded by the $y$-axis and the line with slope $S_1$, as proven in Theorem \ref{['21012507']}. To extend the claim further to the shaded region above, we apply a change of variables and reduce it to the previous case. Please see the proof of Theorem \ref{['2101280']}.
• Figure 5:

Theorems & Definitions (30)

• Theorem 1.1: Hodgson
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• Remark 2.6