Geometric exponents and Kleinian groups

Suppose Λ is the limit set of an analytically finite Kleinian group and that {Ω_j} is an enumeration of the components of Ω = S² \ Λ. Then Σ diam(Ω_j)² < ∞. j This had been conjectured by Maskit. We also define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of Besicovitch and Taylor on the line. Although these exponents may differ for general sets, we show that they are all equal when Λ is the limit set of a non-elementary, analytically finite Kleinian group and they agree with the classical Poincaré exponent.