Ahlfors measure conjecture

In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely generated Kleinian group is either the whole Riemann sphere, or has measure zero.

The conjecture was introduced by Lars Ahlfors,^[1] who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. Richard Canary proved the Ahlfors conjecture for topologically tame groups,^[2] by showing that a topologically tame Kleinian group is geometrically tame, so the Ahlfors conjecture follows from Marden's tameness conjecture that hyperbolic 3-manifolds with finitely generated fundamental groups are topologically tame (homeomorphic to the interior of compact 3-manifolds). This latter conjecture was proved, independently, by Ian Agol^[3] and by Danny Calegari and David Gabai^[4].

Canary also showed that in the case when the limit set is the whole sphere, the action of the Kleinian group on the limit set is ergodic.^[2]