Institute for Mathematical Sciences

Preprint ims94-5

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
  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 <=> dim(Lambda)=2.
  5. If G_n -> 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 the following are equivalent:
  • dim(Lambda(G)) > 1
  • the conical limit set has dimension > 1
  • the Poincare 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.
    View ims94-5 (PostScript format)
    Get TeX source for ims94-5 in tar, zip, or shar format.