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Limit sets of geometrically finite free Kleinian groups. (English) Zbl 0537.30035
It 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)
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