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Hausdorff dimension and Kleinian groups. (English) Zbl 0921.30032
Let \(G\) be a discrete subgroup of \(\text{PSL}_2 (\mathbb{C})\) and consider the action of \(G\) on the 2-sphere \(S^2\) and on the unit ball model \({\mathbf B}\) of three-dimensional hyperbolic space. Denote by \(\Lambda(G)\subset S^2\) the limit set, by \(\Lambda_c(G)\) the conical limit set and by \(\Omega(G)= S^2\setminus \Lambda(G)\) the set of ordinary points of \(G\). \(G\) is named a Kleinian group if \(\Omega(G)\) is non-empty. The Poincaré exponent \(\delta(G)\) is defined to be the abscissa of convergence of the generalized Dirichlet series \(\sum_{G\in G} \exp(-s\rho (0,g(0)))\) where \(\rho\) is the hyperbolic metric on \({\mathbf B}\). \(G\) is called geometrically finite if there exists a finite-sided fundamental polyhedron for the action of \(G\) on \({\mathbf B}\), and \(G\) is called analytically finite if \(\Omega(G)/G\) is a finite union of compact Riemann surfaces with at most finitely many punctures and branch points. The main results of the paper under review are the following:
Theorem 1.1. If \(G\) is non-elementary then \(\delta(G)\) is equal to the Hausdorff dimension of \(\Lambda_c(G)\).
Theorem 1.2. If \(G\) is an analytically finite Kleinian group which is geometrically infinite then the Hausdorff dimension of \(\Lambda(G)\) is equal to 2.
These important theorems were previously proved only under additional restrictions and they have a large number of interesting corollaries. By way of example, the arguments leading to Theorem 1.1 are used to show that \(\delta(G)\) is lower semicontinuous with respect to algebraic convergence if \(G\) has more than one limit point. The authors point out that Theorem 1.1 holds analogously for rank 1 symmetric spaces.
The paper is organized as follows. Section 2 contains the proof of Theorem 1.1 and the lower semicontinuity of \(\delta(G)\). In Section 3, the authors collect some facts about the convex core of the hyperbolic 3-manifold that are needed in later sections. Section 4 contains some facts about the heat kernel and an estimate on Green’s function which is needed in the proof of Theorem 1.2.
In Section 5, the authors prove Theorem 1.2; in fact there are three proofs for the crucial Theorem 5.2. If \(G\) is an analytically finite, geometrically infinite group with \(\delta(G)<2\), then \(\Lambda(G)\) has positive area.
Section 6 is devoted to the proof of Theorem 6.1: If \(G\) is a finitely generated Kleinian group and \((G_n)_{n\geq 1}\) a sequence of Kleinian groups converging algebraically to \(G\) then \(\dim \Lambda(G)\leq \liminf_{n\to\infty} \dim \Lambda(G_n)\) where dim denotes the Hausdorff dimension. In Section 7 the authors analyze \(\dim \Lambda(G)\) as a function on the closure of the Teichmüller space \(T(S)\) of a finite-type hyperbolic surface \(S\). It turns out that \(\dim \Lambda(G)\) is lower semi-continuous on \(\overline{T(S)}\) and continuous everywhere except at the geometrically finite cusps in \(\partial T(A)\) (where it must be discontinuous).
For a fuller account of the authors’ results the reader is advised to consult the first section of the work under review.

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F60 Teichmüller theory for Riemann surfaces
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
22E40 Discrete subgroups of Lie groups
28A78 Hausdorff and packing measures
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