Short Geodesics and End Invariants

Even topologically simple hyperbolic 3-manifolds can have very intricate geometry. Consider in particular a closed surface S of genus 2 or more, and the product N = S×R. This 3-manifold admits a large family of complete, infinite-volume hyperbolic metrics, corresponding to faithful representations ρ : π1(S) → PSL2(C) with discrete image. The geometries of N are very different from the product structure that its topology would suggest. Typically, N contains a complicated pattern of “thin” and “thick” parts. The thin parts are collar neighborhoods of very short geodesics, typically infinitely many. Each one, called a “Margulis tube”, has a well-understood shape, but the way in which these are arranged in N , and in particular the identities of the short geodesics as elements of the fundamental group, are still something of a mystery. This issue is closely related to the basic classification conjecture associated with these manifolds, Thurston’s “ending lamination conjecture”. This conjecture states that certain asymptotic invariants of the geometry of N , called ending invariants, in fact determine N completely. (Actually the classification of hyperbolic structures for any manifold with incompressible boundary reduces to this case, by restriction to boundary subgroups.) In this expository paper we will focus on the following question: What information do the ending invariants give about the presence of very short geodesics in the manifold? We will summarize and discuss the theorem below, part of whose proof appears in [40] and part of which will be in [33], as well as a few conjectures.