Problem 156

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For a Kleinian group $\Gamma$, is the Hausdorff dimension of $\Lambda (\Gamma) <2$ if $\Lambda(\Gamma)$ is not the whole sphere?

(Note, this problem was added by the editor to the end of Rufus' last paper [1])


  1. [bowen1979hausdorff] Bowen R.  1979.  Hausdorff dimension of quasi-circles. Publications Mathématiques de l'IHÉS. 50:11–25.




for a finitely generated kleinian group one knows[ I  heard this much after i left the area] that delta < 2 implies geometrically finite which implies the result....see chris bishop et al...the  analysis in the geometrically finite case is in my papers ACTA...1980s

without the finite generation it is very false by  direct and simple constructions in the classical book by R. Ford "Automorphic Forms"

in my paper for kuipers 60 th birthday I produced a [socalled degenerate] limit of quasi fuchsian groups which had a limit set of  hausdorf dim two and lebesgue measure zero , but positive hausdorf measure for a slightly bigger gauge function with a heuristic argument that rr logloglogr worked...which i recall was later proved by curt mcmullen.

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