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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
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: 251–254, ISSN 0027-8424, JSTOR 57511, MR0194970
Calegari, Danny; Gabai, David (2006), "Shrinkwrapping and the taming of hyperbolic 3-manifolds", Journal of the American Mathematical Society 19 (2): 385–446, doi:10.1090/S0894-0347-05-00513-8, ISSN 0894-0347, MR2188131
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, MR1166330

Mathematics Encyclopedia