Institute for Mathematical Sciences
C. Bishop & P. Jones
Hausdorff dimension and Kleinian groups
Abstract: 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 Poincare 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
The proof also shows that the following are equivalent:
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.
- delta(G) = dim(Lambda_c).
- A simply connected component Omega is either a disk or dim(Omega) >1.
- Lambda(G) is either totally disconnected, a circle or has dimension > 1,
- G is geometrically infinite <=> dim(Lambda)=2.
- If G_n -> G algebraically then dim(Lambda) <= liminf dim(Lambda_n).
- The Minkowski dimension of Lambda equals the Hausdorff dimension.
- If Area(Lambda)=0 then delta(G) = dim(Lambda(G)).
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