Hyperbolic Dehn filling, volume, and transcendentality

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.