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

1. Ahlfors, Lars V. (February 1966). "Fundamental polyhedrons and limit point sets of Kleinian groups". Proceedings of the National Academy of Sciences. 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.
2. 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. JSTOR   2152793. MR   1166330 . Retrieved 2025-08-22.
3. Agol, Ian (2004-05-29), Tameness of hyperbolic 3-manifolds, arXiv, arXiv: math/0405568 , Bibcode:2004math......5568A, doi:10.48550/arXiv.math/0405568 , retrieved 2025-08-22
4. 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. arXiv: math/0407161 . doi:10.1090/S0894-0347-05-00513-8. ISSN   0894-0347. MR   2188131. S2CID   1053364 . Retrieved 2025-08-22.