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.


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


Zbl 0439.30032
Full Text: DOI EuDML


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