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Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\cong \partial \mathbb H^3$, or has zero area. (This result had first been conjectured by Ahlfors, hence the name.)

Are there any known counterexamples to the corresponding result for higher dimensions? What if we add the tameness restriction that $\mathbb{H^n}/\Gamma$ has a manifold compactification?

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  • $\begingroup$ Terra incognita.... $\endgroup$
    – Moishe Kohan
    Commented Jun 11, 2022 at 2:14
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    $\begingroup$ See the discussion at the end of section 10.1 of [M. Kapovich, Kleinian groups in higher dimensions. In "Geometry and Dynamics of Groups and Spaces. In memory of Alexander Reznikov'', M.Kapranov et al (eds). Birkhauser, Progress in Mathematics, Vol. 265, 2007, p. 485-562], math.ucdavis.edu/~kapovich/EPR/klein.pdf. It mentions a complex hyperbolic counterexample. $\endgroup$
    – Igor Belegradek
    Commented Jun 11, 2022 at 22:10
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    $\begingroup$ @IgorBelegradek: +1 for a reference, but, no, you are mistaken. $\endgroup$
    – Moishe Kohan
    Commented Jun 13, 2022 at 4:35
  • $\begingroup$ @MoisheKohan: Oops, I did not read the body question carefully. The complex hyperbolic counterexample is to ergodicity, not the claim that the limit set is the whole sphere at infinity. $\endgroup$
    – Igor Belegradek
    Commented Jun 13, 2022 at 11:16

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