New parameters for Fuchsian groups of genus

Author:
Bernard Maskit

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3643-3652

MSC (1991):
Primary 30F10; Secondary 32G15

Published electronically:
May 13, 1999

MathSciNet review:
1616641

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a new real-analytic embedding of the Teichmüller space of closed Riemann surfaces of genus 2 into . The parameters are explicitly defined in terms of the underlying hyperbolic geometry. The embedding is accomplished by writing down four matrices in , where the entries in these matrices are explicit algebraic functions of the parameters. Explicit inequalities are given to define the image of the embedding; the four matrices corresponding to a point in this image generate a fuchsian group representing a closed Riemann surface of genus .

**1.**J. Gilman and B. Maskit,*An algorithm for 2-generator Fuchsian groups*, Michigan Math. J.**38**(1991), no. 1, 13–32. MR**1091506**, 10.1307/mmj/1029004258**2.**Andrew Haas and Perry Susskind,*The geometry of the hyperelliptic involution in genus two*, Proc. Amer. Math. Soc.**105**(1989), no. 1, 159–165. MR**930247**, 10.1090/S0002-9939-1989-0930247-2**3.**Bernard Maskit,*Explicit matrices for Fuchsian groups*, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 451–466. MR**1292919**, 10.1090/conm/169/01674**4.**Bernard Maskit,*A picture of moduli space*, Invent. Math.**126**(1996), no. 2, 341–390. MR**1411137**, 10.1007/s002220050103

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Additional Information

**Bernard Maskit**

Affiliation:
Department of Mathematics, The University at Stony Brook, Stony Brook, New York 11794-3651

Email:
bernie@math.sunysb.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04973-4

Received by editor(s):
October 20, 1997

Received by editor(s) in revised form:
February 20, 1998

Published electronically:
May 13, 1999

Additional Notes:
Research supported in part by NSF Grant DMS 9500557.

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1999
American Mathematical Society