## The multiplier-series of a Schottky group.(English)Zbl 0869.30036

Let $$g\geq 1$$. A Schottky group of rank $$g$$ is a geometrically finite discrete subgroup of $$\text{PSL}_2(C)$$ which is a free subgroup of rank $$g$$ and which is purely hyperbolic. The Schottky space of genus $$g$$, $$S_g$$, is the set of all conjugacy classes of faithful representations of the free group $$\Gamma_g$$ of rank $$g$$ in $$\text{PSL}_2(C)$$ whose image is a Schottky group. There is a complex structure on $$S_g$$, and this space can be realized as an open domain in $$C^{3g-3}$$. Moreover, there is a canonical action of the outer automorphism group $$\Gamma_g$$ on $$S_g$$, and this action is bianalytic and properly discontinuous (the author refers to one of his earlier papers for these facts). The main result of this paper is the following. Let $$S$$ be a Schottky group. Then the series of weighted multipliers $$\lambda(\gamma)$$ taken over all conjugacy classes of elements $$\gamma$$ in $$S$$ converges absolutely. Moreover, this convergence is locally absolutely uniform on the Schottky space. The author obtains therefore a family of holomorphic functions on the quotient space $$R_g$$ of $$S_g$$ by $$\text{Out}(\Gamma_g)$$. The convergence result is a generalization of a result stated by R. Bowen in his paper “Hausdorff dimension of quasi-circles” [Publ. Math. Inst. Hautes Etud. Sci. 50, 11-25 (1979; Zbl 0439.30032)]. The proof given here is different from the one which has been outlined by Bowen.

### MSC:

 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 28D05 Measure-preserving transformations

Zbl 0439.30032
Full Text:

### References:

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