zbMATH — the first resource for mathematics

Approximation of a map between one-dimensional Teichmüller spaces. (English) Zbl 1091.30043
The Teichmüller space of once-punctured tori can be realized as the upper half-plane $$\mathbb H$$, or via the Maskit embedding as a proper subset of $$\mathbb H$$. We construct and approximate the explicit biholomorphic map from Maskit’s embedding to $$\mathbb H$$. This map involves the integration of an abelian differential constructed using an infinite sum over the elements of a Kleinian group. We approximate this sum and thereby find the locations of the square torus and the hexagonal torus in Maskit’s embedding, and we show that the biholomorphism does not map vertical pleating rays in Maskit’s embedding on vertical lines in $$\mathbb H$$.

MSC:
 30F60 Teichmüller theory for Riemann surfaces
Full Text:
References:
 [1] Farkas H. M., Riemann surfaces,, 2. ed. (1991) · Zbl 1078.14522 [2] Keen L., Topology 32 (4) pp 719– (1993) · Zbl 0794.30037 [3] Keen L., Proc. London Math. Soc. (3) 69 (1) pp 72– (1994) · Zbl 0807.30031 [4] King J. T., Introduction to numerical computation (1984) [5] Kra I., Automorphic forms and Kleinian groups (1972) · Zbl 0253.30015 [6] Kra I., J. Analyse Math. 43 pp 51– (1984) · Zbl 0569.30037 [7] DOI: 10.1007/BF02392375 · Zbl 0543.30037 [8] Kra I., Ann. Polon. Math. 46 pp 147– (1985) [9] Kra I., Holomorphic functions and moduli II (Berkeley, 1986) pp 221– (1988) [10] Kra I., J. Amer. Math. Soc. 3 pp 499– (1990) [11] Kra I., Proc. Amer. Math. Soc. 111 (3) pp 803– (1991) [12] Lehner J., Discontinuous groups and automorphic functions (1964) · Zbl 0178.42902 [13] Lepowsky J., Notices Amer. Math. Soc. 46 (1) pp 17– (1999) [14] DOI: 10.1090/S0002-9904-1974-13600-1 · Zbl 0292.30016 [15] DOI: 10.1007/978-3-642-61590-0 [16] Matsuzaki K., Hyperbolic manifolds and Kleinian groups (1998) · Zbl 0892.30035 [17] DOI: 10.2307/2944328 · Zbl 0718.30033 [18] McMullen, C. ”Rational Maps and Kleinian Groups”. Proceedings of the International Congress of Mathematicians. 1990, Kyoto. Edited by: Satake, I. vol. II, pp.889–899. New York: Springer. [McMullen 1991b] · Zbl 0764.30022 [19] McMullen C., J. Amer. Math. Soc. 11 (2) pp 283– (1998) · Zbl 0890.30031 [20] Springer G., Introduction to Riemann Surfaces (1957) · Zbl 0078.06602 [21] Swinnerton-Dyer H. P. F., Analytic theory of Abelian varieties (1974) · Zbl 0299.14021 [22] Wright D. J., ”The shape of the boundary of the Teichmüller space of once-punctured tori in Maskit’s embedding”
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.