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Hyperbolic structures for surfaces of infinite type. (English) Zbl 0768.30028

The author considers hyperbolic surfaces of infinite type, that are surfaces with an infinitely generated fundamental group. In particular he constructs surfaces called (tight) flute spaces from a union of pairs of (tight) pants each glued to the next along a common boundary geodesic. The author showed that given any sequence of positive numbers \(\{d_ i\}\) there exists a hyperbolic structure for a flute space having a nested sequence of geodesics of hyperbolic distances \(\{d_ i\}\).
Secondly he supplied necessary and sufficient conditions in the form of inequalities involving distances between geodesics for the existence of a hyperbolic structure on a tight flute. It has been shown that a Fuchsian group associated to a tight flute is of the first kind if \(\sum d_ i=\infty\). The author supplies sufficient conditions on an infinite number of polygons so that when glued together they form a new fundamental polygon. Finally, a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined.

MSC:

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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References:

[1] William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. · Zbl 0452.32015
[2] Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. · Zbl 0196.33801
[3] A. Basmajian, Hyperbolic invariants for infinitely generated Fuchsian groups, Ph.D. thesis, Stony Brook University, 1987.
[4] Ara Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 65 – 74. · Zbl 0673.30032
[5] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. · Zbl 0528.30001
[6] Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. · Zbl 0629.30002
[7] Linda Keen, On infinitely generated Fuchsian groups, J. Indian Math. Soc. (N.S.) 35 (1971), 67 – 85 (1972). · Zbl 0251.20053
[8] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. · Zbl 0627.30039
[9] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603
[10] Norman Purzitsky, A cutting and pasting of noncompact polygons with applications to Fuchsian groups, Acta Math. 143 (1979), no. 3-4, 233 – 250. · Zbl 0427.30039
[11] -, Fricke polygons for infinitely generated fuchsian groups. I, preprint.
[12] W. Thurston, The geometry and topology of \( 3\)-manifolds, Lecture Notes, 1977.
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