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 0.
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The conjecture was introduced by Ahlfors (1966), who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. Canary (1993) proved the Ahlfors conjecture for topologically tame groups, 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 Agol (2004) and by Calegari & Gabai (2006).
Canary (1993) 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.
