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The Hausdorff dimension of the limit set of a geometrically finite Kleinian group. (English) Zbl 0539.30034
Let G be a geometrically finite discrete group of isometries of \((n+1)\)- dimensional hyperbolic space. Let \(L(G)\) be the limit set of G, \(\dim_ HL(G)\) its Hausdorff dimension and \(\delta(G)\) the ”exponential of convergence” of G. It is known, by an argument going back to Beardon, that \(\dim_ HL(G)\leq \delta(G).\) There is reason to believe that the opposite inequality holds without any restriction on G. This at least is true when \(n=1\); when \(n=2\) and G is finitely generated it has been proved by Sullivan and it appears that his argument shows that it is true for all geometrically finite groups with no restrictin on n. These results require fairly deep methods which also show that when \(L(G)\) is not the entire boundary of hyperbolic space then \(\delta(G)<n.\) In this paper a clear and elementary proof is given of this latter inequality (and of the corresponding statement about the Hausdorff dimension). The argument is reminiscent of that given by A. F. Beardon [Acta Math. 127, 221-258 (1971; Zbl 0235.30022)] but is not shorter and simpler.
Reviewer: S.J.Patterson

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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