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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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