Ahlfors measure conjecture

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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 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. 1 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). "Tameness of hyperbolic 3-manifolds". arXiv: math/0405568 .
  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.


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