Pleating coordinates for the Teichmüller space of a punctured torus

Authors:
Linda Keen and Caroline Series

Journal:
Bull. Amer. Math. Soc. **26** (1992), 141-146

MSC (2000):
Primary 30F40; Secondary 30F60, 32G15, 57N05, 57S30

MathSciNet review:
1110439

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Abstract: We construct new coordinates for the Teichmüller space Teich of a punctured torus into . The coordinates depend on the representation of Teich as a space of marked Kleinian groups that depend holomorphically on a parameter varying in a simply connected domain in **C**. They describe the geometry of the hyperbolic manifold ; they reflect exactly the visual patterns one sees in the limit sets of the groups ; and they are directly computable from the generators of .

**[1]**D. B. A. Epstein and A. Marden,*Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces*, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR**903852****[2]**Linda Keen, Bernard Maskit, and Caroline Series,*Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets*, J. Reine Angew. Math.**436**(1993), 209–219. MR**1207287****[3]**Linda Keen and Caroline Series,*Continuity of convex hull boundaries*, Pacific J. Math.**168**(1995), no. 1, 183–206. MR**1331998****[4]**-,*Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori*, IMS SUNY, 1991/2.**[5]**-,*The Riley slice of Shottky space*, Warwick Univ., preprint, 1991.**[6]**Curt McMullen,*Cusps are dense*, Ann. of Math. (2)**133**(1991), no. 1, 217–247. MR**1087348**, 10.2307/2944328**[7]**-, personal communication.**[8]**Caroline Series,*The geometry of Markoff numbers*, Math. Intelligencer**7**(1985), no. 3, 20–29. MR**795536**, 10.1007/BF03025802**[9]**W. P. Thurston,*Geometry and topology of three manifolds*, Lecture notes, Princeton Univ., NJ, 1979.

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DOI:
http://dx.doi.org/10.1090/S0273-0979-1992-00259-8

Article copyright:
© Copyright 1992
American Mathematical Society