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- W4289920008 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 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." @default.
- W4289920008 created "2022-08-06" @default.
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- W4289920008 creator A5089476171 @default.
- W4289920008 date "1994-03-21" @default.
- W4289920008 modified "2023-10-17" @default.
- W4289920008 title "Hausdorff dimension and Kleinian groups" @default.
- W4289920008 doi "https://doi.org/10.48550/arxiv.math/9403222" @default.
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