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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
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Mathematical Citation Quotient (MCQ) 2018: 0.53

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1615-715X
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Exceptional points for finitely generated Fuchsian groups of the first kind

Joseph Fera / Andrew Lazowski
  • Corresponding author
  • Department of Mathematics, Sacred Heart University, 5151 Park Avenue, Fairfield, CT 06825, USA
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Published Online: 2019-06-30 | DOI: https://doi.org/10.1515/advgeom-2019-0013

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.

Keywords: Fuchsian groups; Dirichlet regions

MSC 2010: Primary 20H10; Secondary 30F10

References

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

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

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

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    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 0593.30046CrossrefGoogle Scholar

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

About the article

Received: 2018-09-06

Revised: 2018-09-20

Published Online: 2019-06-30


Communicated by: J. Ratcliffe


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2019-0013.

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