Limit sets of geometrically finite free Kleinian groups.

*(English)*Zbl 0537.30035It is known that the Hausdorff dimension of the limit set of a geometrically finite Kleinian group is equal to its exponent of convergence. Historically the first approach towards a proof of this statement, and also the most elementary one, was found by T. Akaza. It has the disadvantage that it appears to be necessary to make some further assumptions on the nature of the group; at any rate such assumptions have had to be made in all applications of the method to date. The purpose of this paper is to give a description of the method applied to a fairly general class of groups, namely those that are free as groups. Although the authors imply in § 2.2 that parabolic elements are permitted it is known that Proposition 5 of this paper is false in this case [see D. Sullivan, Entropy, Hausdorff measure old and new, and limit sets of geometrically finite Kleinian groups (to appear in Acta Math.); Theorem 2). The reason appears to be that Proposition 1 is not correctly formulated; were this done then it would presumably follow that the main result of the paper would hold under the additional assumption that the group contains no parabolic elements.

Reviewer: S.J.Patterson

##### MSC:

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

##### Keywords:

Poincaré dimension; Hausdorff dimension; limit set; exponent of convergence; parabolic elements
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\textit{T. Akaza} and \textit{K. Inoue}, Tohoku Math. J. (2) 36, 1--16 (1984; Zbl 0537.30035)

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##### References:

[1] | T. AKAZA, Local properties of the singular sets of some Kleinian groups, Thoku Math. J. 25 (1973), 1-22. · Zbl 0264.30025 |

[2] | T. AKAZA AND K. INOUE, On the limit set of a geometrically finite Kleinian group, Sci Rep. of Kanazawa Univ. 27 (1983), 85-116. · Zbl 0514.30034 |

[3] | A. F. BEARDON AND P. J. NICHOLLS, On classical series associated with Kleinian groups, J. London Math. Soc.5 (1972), 645-655. · Zbl 0246.30018 |

[4] | A. F. BEARDON AND B. MASKIT, Limit points of Kleinian groups and finite-sided funda mental polyhedra, Acta Math, 132 (1974), 1-12. · Zbl 0277.30017 |

[5] | L. R. FORD, Automorphic functions, 2nd ed., Chelsea, New York, 1951 · JFM 55.0810.04 |

[6] | S. J. PATTERSON, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273 · Zbl 0336.30005 |

[7] | D. SULLIVAN, The density at infinity of a discrete group of hyperbolic motions, Inst Hautes Etudes Sci.Publ. Math. 50 (1979), 171-202. · Zbl 0439.30034 |

[8] | D. SULLIVAN, Discrete conformal groups and measurable dynamics, Bull, of Amer. Math Soc. 6 (1982), 57-73. · Zbl 0489.58027 |

[9] | D. SULLIVAN, Entropy, Hausdorff measure old and new, and limit sets of geometricall finite Kleinian groups, to appear in Acta Math. · Zbl 0566.58022 |

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