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.

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.

References

• Agol, Ian (2004), Tameness of hyperbolic 3-manifolds, arXiv:math/0405568, Bibcode:2004math......5568A
• Ahlfors, Lars V. (1966), "Fundamental polyhedrons and limit point sets of Kleinian groups", Proceedings of the National Academy of Sciences of the United States of America, 55 (2): 251–254, Bibcode:1966PNAS...55..251A, doi:10.1073/pnas.55.2.251, ISSN 0027-8424, JSTOR 57511, MR 0194970, PMC 224131, PMID 16591331
• Calegari, Danny; Gabai, David (2006), "Shrinkwrapping and the taming of hyperbolic 3-manifolds", Journal of the American Mathematical Society, 19 (2): 385–446, arXiv:math/0407161, doi:10.1090/S0894-0347-05-00513-8, ISSN 0894-0347, MR 2188131, S2CID 1053364
• Canary, Richard D. (1993), "Ends of hyperbolic 3-manifolds", Journal of the American Mathematical Society, 6 (1): 1–35, doi:10.2307/2152793, ISSN 0894-0347, JSTOR 2152793, MR 1166330