Hyperbolic Dehn filling, volume, and transcendentality

TL;DR

This paper investigates the distribution of volumes of Dehn fillings on 1-cusped hyperbolic 3-manifolds, combining computational experiments with theoretical analysis to understand the growth rate of the number of fillings with a given volume.

Contribution

It introduces a new theoretical framework for understanding the behavior of Dehn fillings' volumes and proves that their count grows slower than any polynomial rate.

Findings

Estimated the number of Dehn fillings for various volumes through computational experiments.

Proved that the growth of the number of fillings is slower than any polynomial function of the filling coefficient.

Provided insights into the transcendental nature of volume distributions in hyperbolic 3-manifolds.

Abstract

Let $M$ be a 1-cusped hyperbolic 3-manifold. In this paper, we study the behavior of $N_M(v)$, the number of Dehn fillings of $M$ with a given volume $v(\in \mathbb{R})$. We conduct extensive computational experiments to estimate $N_M$ and propose a theoretical framework to explain its behavior. Further, we prove that the growth of $N_M$ is slower than any power of its filling coefficient.