Exceptional points for finitely generated Fuchsian groups of the first kind

Joseph Fera 1  and Andrew Lazowski 2
  • 1 Department of Mathematics, Lehman College CUNY, 250 Beadford Park Boulevard West, Bronx, USA
  • 2 Department of Mathematics, Sacred Heart University, 5151 Park Avenue, Fairfield, USA
Joseph Fera
  • Department of Mathematics, Lehman College CUNY, 250 Beadford Park Boulevard West, Bronx, NY 10468, USA
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and Andrew Lazowski
  • Corresponding author
  • Department of Mathematics, Sacred Heart University, 5151 Park Avenue, Fairfield, CT 06825, USA
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Abstract

Let G be a finitely generated Fuchsian group of the first kind and let (g : m1, m2, …, mn) be its shortened signature. Beardon showed that almost every Dirichlet region for G has 12g + 4n − 6 sides. Points in ℍ corresponding to Dirichlet regions for G with fewer sides are called exceptional for G. We generalize previously established methods to show that, for any such G, its set of exceptional points is uncountable.

  • [1]

    A. F. Beardon, The geometry of discrete groups. Springer 1983. MR698777 Zbl 0528.30001

  • [2]

    J. Fera, Exceptional points for cocompact Fuchsian groups. Ann. Acad. Sci. Fenn. Math. 39 (2014), 463–472. MR3186824 Zbl 1296.30050

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  • [3]

    S. Katok, Fuchsian groups. University of Chicago Press, Chicago, IL 1992. MR1177168 Zbl 0753.30001

  • [4]

    M. Näätänen, On the stability of identification patterns for Dirichlet regions. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 411–417. MR802503 Zbl 0593.30046

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    • Export Citation
  • [5]

    J. G. Ratcliffe, Foundations of hyperbolic manifolds. Springer 2006. MR2249478 Zbl 1106.51009

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.

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