A large and important class of hyperbolic 3-manifolds arise as the complements of links in the 3-sphere. In this talk, I address the question of whether there can be a sequence of link complements with volumes tending to infinity that form an expanding family, in the sense that their Cheeger constants are bounded away from zero.
I show that the answer is no if the links are all alternating or highly twisted. The proof uses methods from hyperbolic 3-manifold theory, and, in addition, Lipton and Tarjan's theorem on separators for planar graphs. Some consequences and extensions will be given, which have implications for surgery diagrams for 3-manifolds.