Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A bound for the number of automorphisms of a compact Klein surface with boundary


Author: Coy L. May
Journal: Proc. Amer. Math. Soc. 63 (1977), 273-280
MSC: Primary 30A46; Secondary 14H30, 32L05
DOI: https://doi.org/10.1090/S0002-9939-1977-0435385-0
MathSciNet review: 0435385
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For an integer $ g \geqslant 2$, let $ \nu (g)$ be the order of the largest group of automorphisms of a compact Klein surface with nonempty boundary of genus g. We show that $ \nu (g) \geqslant 4(g + 1)$ for all g and that for an infinite number of values of $ \nu (g) = 4(g + 1)$.


References [Enhancements On Off] (What's this?)

  • [1] R. D. M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398-408. MR 36 #5333. MR 0222281 (36:5333)
  • [2] N. L. Alling and N. Greenleaf, Foundations of the theory of Klein surfaces, Springer-Verlag, Berlin and New York, 1971. MR 0333163 (48:11488)
  • [3] M. Heins, On the number of $ 1{\text{-}}1$ directly conformal maps which a multiply-connected plane region of finite connectivity $ p\;( > 2)$ admits onto itself, Bull. Amer. Math. Soc. 52 (1946), 454-457. MR 8, 21. MR 0016469 (8:21d)
  • [4] A. Hurwitz, Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403-442. MR 1510753
  • [5] Takao Kato, On the number of automorphisms of a compact bordered Riemann surface, Kōdai Math. Sem. Rep. 24 (1972), 224-233. MR 46 #5610. MR 0306484 (46:5610)
  • [6] C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44 (1969), 265-272. MR 38 #4674. MR 0236378 (38:4674)
  • [7] C. L. May, Automorphisms of compact Klein surfaces with boundary, Pacific J. Math. 59 (1975), 199-210. MR 0399451 (53:3295)
  • [8] -, Large automorphism groups of compact Klein surfaces with boundary. I, Glasgow Math. J. 18 (1977), 1-10. MR 0425113 (54:13071)
  • [9] -, Cyclic automorphism groups of compact Klein surfaces with boundary (to appear).
  • [10] K. Oikawa, Notes on conformal mappings of a Riemann surface onto itself, Kōdai Math. Sem. Rep. 8 (1956), 23-30; A supplement to ``Notes on conformal mappings of a Riemann surface onto itself", ibid. 8 (1956), 115-116. MR 18, 290; 797. MR 0080730 (18:290d)
  • [11] W. R. Scott, Group theory, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 29 #4785. MR 0167513 (29:4785)
  • [12] R. Tsuji, On conformal mapping of a hyperelliptic Riemann surface onto itself, Kōdai Sem. Rep. 10 (1958), 127-136. MR 20 #6521. MR 0100085 (20:6521)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A46, 14H30, 32L05

Retrieve articles in all journals with MSC: 30A46, 14H30, 32L05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0435385-0
Keywords: Klein surface, automorphism group, algebraic genus, complex double, Riemann surface, ramified covering, Hurwitz ramification formula
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society