# On the number of hyperbolic Dehn fillings of a given volume

## TL;DR

This paper proves that for a specific class of hyperbolic 3-manifolds, the number of hyperbolic Dehn fillings with a fixed volume is uniformly bounded, revealing a new constraint on their geometric structures.

## Contribution

It establishes a uniform bound on the number of hyperbolic Dehn fillings with a given volume for 1-cusped hyperbolic 3-manifolds with quadratic cusp shape.

## Key findings

- Bound c(M) depends on the manifold M
- Number of fillings with fixed volume is uniformly limited
- Results apply to manifolds with quadratic cusp shape

## Abstract

Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is quadratic. We show that there exists c=c(M) such that the number of hyperbolic Dehn fillings of M with any given volume v is uniformly bounded by c.