Word length and limit sets of Kleinian groups. (Q2574408)

If \(\Gamma\) is a finitely generated group of isometries of a real or complex hyperbolic space then it is meaningful to compare the combinatorial metric on the Cayley graph of \(\Gamma\) with the hyperbolic metric induced by mapping \(\Gamma\) to an orbit in the hyperbolic space. The first results of this type (for spaces of arbitrary dimension) were obtained in 1965 by \textit{M. Eichler} [Acta Arith. 11, 169-180 (1965; Zbl 0148.32503)]. These results were put in a more general geometric context by \textit{W. J. Floyd} [Invent. Math. 57, 205-218 (1980; Zbl 0428.20022)] in 1980 and \textit{J. W. Cannon} [Geom. Dedicata 16, 123-148 (1984; Zbl 0606.57003)] in 1984. In this paper the author starts from Floyd's work and develops it to prove the following elegant theorem: Let \[ \alpha(\Gamma)=\sup\Bigl\{\alpha:\sup_{g\in\Gamma} \tfrac{|g|^\alpha}{e^{d_h(o,g(o))}}<\infty\Bigr\}. \] Here \(d_h\) denotes the hyperbolic distance, \(o\) is a point of the hyperbolic space and \(|g|\) denotes the minimal word length of \(g\) with respect to a fixed (finite) set of generators. Then, if \(\Gamma\) is as above one has that the following statements are equivalent: (i) \(\Gamma\) is convex co-compact, (ii) \(\alpha(\Gamma)>2\) and (iii) \(\alpha(\Gamma)=\infty\). One knew before this (Floyd, loc. cit.) that if \(\Gamma\) were geometrically finite then either \(\alpha(\Gamma)=2\) or \(\alpha=\infty\) according to whether or not \(\Gamma\) has parabolic elements.