Hyperbolic Dehn surgery

In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

Such an operation is often also called hyperbolic Dehn filling, as Dehn surgery proper refers to a "drill and fill" operation on a link which consists of drilling out a neighborhood of the link and then filling back in with solid tori. Hyperbolic Dehn surgery actually only involves "filling".

We will generally assume that a hyperbolic 3-manifold is complete.

Suppose M is a cusped hyperbolic 3-manifold with n cusps. M can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. Let ${\displaystyle M(u_{1},u_{2},\dots ,u_{n})}$ denote the manifold obtained from M by filling in the i-th boundary torus with a solid torus using the slope ${\displaystyle u_{i}=p_{i}/q_{i}}$ where each pair ${\displaystyle p_{i}}$ and ${\displaystyle q_{i}}$ are coprime integers. We allow a ${\displaystyle u_{i}}$ to be ${\displaystyle \infty }$ which means we do not fill in that cusp, i.e. do the "empty" Dehn filling. So M = ${\displaystyle M(\infty ,\dots ,\infty )}$.

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