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## User: peter-jones

### Title: Hausdorff dimension and Kleinian groups

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$. (3) Lambda(G) is either totally disconnected, a circle or has dimension > 1, (4) G is geometrically infinite iff dim(Lambda)=2. (5) If G_n \to G algebraically then dim(Lambda) <= \liminf dim(Lambda_n). (6) The Minkowski dimension of Lambda equals the Hausdorff dimension. (7) If Area(Lambda)=0 then delta(G) = dim(Lambda(G)). The proof also shows that \dim(Lambda(G)) > 1 iff the conical limit set has dimension > 1 iff the Poincar\'e exponent of the group is > 1. Furthermore, a simply connected component of Omega(G) either is a disk or has non-differentiable boundary in the the sense that the (inner) tangent points of \partial Omega have zero 1-dimensional measure. Almost every point (with respect to harmonic measure) is a twist point. Click here to read this post out ID: 129821; Unique Viewers: 0 Unique Voters: 0 Latest Change: Nov. 30, 2020, 2:42 a.m. Changes: Dictionaries: Words: Spaces: Comments: Newcom ### Posts: Total post views: 247636 Sort: <div class="abstract-content selected" id="enc-abstract"> <p> I met a traveler from an antique landWho said:… Words: Votes: Views: 0 Latest: June 6, 2022, 3:06 a.m. Positron annihilation lifetime spectroscopy (PALS) and positron-electron momentum density (PEMD) studies on multilayers of PbSe nanocrystals (N… Words: Votes: Views: 0 Latest: April 18, 2021, 9:38 p.m. Air pollution has long been a serious environmental health challenge, especially in metropolitan cities, where air pollutant concentrations are… Words: Votes: Views: 0 Latest: April 18, 2021, 8:51 p.m. A connected ordering$(v_1, v_2, \ldots, v_n)$of$V(G)$is an ordering of the vertices such that$v_i$has at least one neighbour in$\{v_1, \…
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