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]
References
- ↑ Ahlfors, Lars V. (February 1966). "Fundamental polyhedrons and limit point sets of Kleinian groups" (in en). Proceedings of the National Academy of Sciences 55 (2): 251–254. doi:10.1073/pnas.55.2.251. ISSN 0027-8424. PMID 16591331. Bibcode: 1966PNAS...55..251A.
- ↑ 2.0 2.1 Canary, Richard D. (1993). "Ends of hyperbolic 3-manifolds". Journal of the American Mathematical Society 6 (1): 1–35. doi:10.1090/S0894-0347-1993-1166330-8. ISSN 0894-0347. https://www.ams.org/jams/1993-06-01/S0894-0347-1993-1166330-8/. Retrieved 2025-08-22.
- ↑ Agol, Ian (2004). "Tameness of hyperbolic 3-manifolds". arXiv:math/0405568.
- ↑ Calegari, Danny; Gabai, David (2006-04-01). "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. https://www.ams.org/jams/2006-19-02/S0894-0347-05-00513-8/. Retrieved 2025-08-22.
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