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Let G be a non-elementary, finitely generated Kleinian group, Lambda(G) its
limit set and Omega(G) = S \ Lambda(G) (S = the sphere) its set of
discontinuity. Let delta(G) be the critical exponent for the Poincar\'e series
and let Lambda_c be the conical limit set of G. Suppose Omega_0 is a simply
connected component of Omega(G). We prove that
(1) delta(G) = dim(Lambda_c).
(2) A simply connected component Omega is either a disk or dim(Omega)>1$.
(3) Lambda(G) is either totally disconnected, a circle or has dimension > 1,
(4) G is geometrically infinite iff dim(Lambda)=2.
(5) If G_n \to G algebraically then dim(Lambda) <= \liminf dim(Lambda_n).
(6) The Minkowski dimension of Lambda equals the Hausdorff dimension.
(7) If Area(Lambda)=0 then delta(G) = dim(Lambda(G)).
The proof also shows that \dim(Lambda(G)) > 1 iff the conical limit set has
dimension > 1 iff the Poincar\'e exponent of the group is > 1. Furthermore, a
simply connected component of Omega(G) either is a disk or has
non-differentiable boundary in the the sense that the (inner) tangent points of
\partial Omega have zero 1-dimensional measure. Almost every point (with
respect to harmonic measure) is a twist point.

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