×

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

Citations:

Zbl 0439.30032
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Bowen, R.: Hausdorff dimension of quasi – circles. I.H.E.S. Publ. Math.50, 1979, 11–26 · Zbl 0439.30032
[2] Büser, J.: Parabolic completion of Schottky space. Preprint Univ. Bochum, 1994
[3] Freitag, E.: Siegelsche Modulfunktionen. Springer, New York Heidelberg Berlin, 1983 · Zbl 0498.10016
[4] Gerritzen, L., Herrlich, F.: The extended Schottky space. J. reine angew. Math.389 (1988), 190–208 · Zbl 0639.30040
[5] Herrlich, F.: The extended Teichmüller space. Math. Z.203 (1990), 279–291 · Zbl 0662.32021
[6] Lehner, J.: Discontinuous groups and automorphic functions. Math. Surveys VIII. AMS, Providence, 1964 · Zbl 0178.42902
[7] Luft, E.: Actions of the homeotopy group of an orientable 3-dimensional handlebody. Math. Ann.234 (1978), 279–292 · Zbl 0364.57011
[8] Manin, Y. I.: Three-dimensional hyperbolic geometry as adic Arakelov geometry. Invent. math.104 (1991), 223–244 · Zbl 0754.14014
[9] Maskit, B.: Kleinian groups. Springer, Berlin et al., 1988 · Zbl 0627.30039
[10] Nag, S.: The complex analytic theory of Teichmüller spaces. Wiley- Interscience, New York, 1988 · Zbl 0667.30040
[11] Namikawa, Y.: Toroidal compactification of Siegel spaces. Springer, New York Heidelberg Berlin, 1980 · Zbl 0466.14011
[12] Perry, P. A.: The Selberg zeta function and a local trace formula for Kleinian groups. J. reine angew. Math.410 (1990) 116–152 · Zbl 0697.10027
[13] Pollicott, M.: Closed geodesics and zeta functions. In: Bedford, T., Keane, M., Series, C. (eds.): Ergodic theory, symbolic dynamics, and hyperbolic geometry. Oxford University Press, Oxford, New York, Tokyo, 1991, 153–174 · Zbl 0753.58028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.