Holomorphic Functions and Moduli II
Volume 11 of the series Mathematical Sciences Research Institute Publications pp 251265
Parameters for Fuchsian Groups I: Signature (0, 4)
 Bernard MaskitAffiliated withDepartment of Mathematics, State University of New York at Stony Brook
Abstract
This is the first of a series of notes presenting new parameters for certain torsionfree finitely generated Fuchsian and quasifuchsian groups. In this note we consider signature (0, 4). Other low signatures, as well as the general case, will be dealt with elsewhere. Every Fuchsian group of signature (0, 4), acting on the upper halfplane ∪, can be generated by four parabolic transformations, A,B,C,D, where the product ABCD = 1. Normalize so that AB has its attracting fixed point at ∞, its repelling fixed point at 0, and so that the fixed point of C is at 1. Let x be the fixed point of D, and let y be the fixed point of B. Then x > 1 and y < 0. We show that x and y serve as parameters for the deformation space of these groups (this is really two results, one having to do with Fuchsian, and the other with quasifuchsian groups). We also explicitly write the matrices A,B,C,D in PGL(2,ℝ) ^{+} (these are 2 × 2 real matrices with positive determinant) as functions of x and y; this gives an explicit example of a stratification (see [KM]). We also construct an explicit fundamental domain for the Teichmüller modular group for signature (0, 4), and we identify the side pairing transformations.
 Title
 Parameters for Fuchsian Groups I: Signature (0, 4)
 Book Title
 Holomorphic Functions and Moduli II
 Book Subtitle
 Proceedings of a Workshop held March 13–19, 1986
 Pages
 pp 251265
 Copyright
 1988
 DOI
 10.1007/9781461396116_17
 Print ISBN
 9781461396130
 Online ISBN
 9781461396116
 Series Title
 Mathematical Sciences Research Institute Publications
 Series Volume
 11
 Series ISSN
 09404740
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag New York Inc.
 Additional Links
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 Editors

 D. Drasin ^{(1)} ^{(6)}
 C. J. Earle ^{(2)} ^{(6)}
 F. W. Gehring ^{(3)} ^{(6)}
 I. Kra ^{(4)} ^{(6)}
 A. Marden ^{(5)} ^{(6)}
 Editor Affiliations

 1. Department of Mathematics, Purdue University
 6. Mathematical Sciences Research Institute
 2. Department of Mathematics, Cornell University
 3. Department of Mathematics, University of Michigan
 4. Department of Mathematics, State University of New York at Stony Brook
 5. Department of Mathematics, University of Minnesota
 Authors

 Bernard Maskit ^{(7)}
 Author Affiliations

 7. Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY, 11794, USA
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