Is it true that the Hausdorff dimension of the limit set of a Kleinian group is the supremum of Hausdorff dimensions of finitely generated subgroups [perhaps under addtional hypotheses?]I can't seem to find a reference which proves (or disproves) this fact. This is related to my previous question on commutator of $\Gamma(2)$...

**EDIT:** It *is* a Theorem of D. Sullivan (1979) that the critical exponent $\delta(\Gamma)$ is the $\sup$ of critical exponents for finitely generated subgroups, and it is another theorem of Sullivan (1984) that the Hausdorff dimension of the *radial* limit set is equal to the critical exponent.