| 词条 | Ahlfors measure conjecture |
| 释义 |
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. The conjecture was introduced by {{harvs|txt|last=Ahlfors|authorlink=Lars Ahlfors|year=1966}}, who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. {{harvtxt|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 {{harvtxt|Agol|2004}} and by {{harvtxt|Calegari|Gabai|2006}}. {{harvtxt|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.References
3 : Kleinian groups|Theorems in analysis|Conjectures that have been proved |
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