Hyperbolic 3-manifolds and the Langlands program

Wojtek Wawrów

Waldhausen's Virtual Haken Conjecture asks whether an arbitrary closed hyperbolic 3-manifold admits a finite cover which admits an embedded "incompressible" surface. As this question remains open, many of its variants have been proposed, including a question of Cooper whether there are 3-manifolds which admit arbitrarily large covers by rational homology spheres.

In a surprising turn of events, this problem, a question from differential geometry, has been resolved by Calegari and Dunfield conditionally on some statements from the Langlands program, a deep web of conjectures connecting number theory and arithmetic geometry.

In this talk we shall describe the tools which come into that proof in order to explain this unexpected connection. No prior familiarity with any notions related to the Langlands program will be assumed.