Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Surface symmetries and $ PSL_2(p)$


Authors: Murad Özaydin, Charlotte Simmons and Jennifer Taback
Journal: Trans. Amer. Math. Soc. 359 (2007), 2243-2268
MSC (2000): Primary 57M60; Secondary 12F12, 20H10, 30F10
DOI: https://doi.org/10.1090/S0002-9947-06-04011-6
Published electronically: December 19, 2006
MathSciNet review: 2276619
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We classify, up to conjugacy, all orientation-preserving actions of $ PSL_2(p)$ on closed connected orientable surfaces with spherical quotients. This classification is valid in the topological, PL, smooth, conformal, geometric and algebraic categories and is related to the Inverse Galois Problem.


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

  • [B] A. F. Beardon, The geometry of discrete groups, Springer-Verlag, New York (1983). MR 0698777 (85d:22026)
  • [D] L.E. Dickson, Linear groups with an exposition of the Galois field theory, Leipzig (1901) reprinted by Dover, New York (1960). MR 104735 (21:3488)
  • [EKS] A.L. Edmonds, R.S. Kulkarni and R. Stong, Realizability of branched coverings of surfaces, Transactions of the Amer. Math. Soc., 282, no. 2 (1984), pp. 773-790. MR 0732119 (85k:57005)
  • [FK] H.M. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag, New York (1980). MR 0583745 (82c:30067)
  • [FGM] D. Frohardt, R. Guralnick, and K. Magaard, Genus 0 Actions of Groups of Lie Rank 1, Proceedings of Symposia in Pure Mathematics, Fried and Ihara, eds., 70 (2002), pp. 449-483. MR 1935417 (2003j:20019)
  • [GS1] H. Glover and D. Sjerve, Representing $ PSL_2(p)$ on a Riemann surface of least genus, L'ensignment Mathematique 31 (1985), pp. 305-325. MR 0819357 (87a:20051)
  • [GS2] H. Glover and D. Sjerve, The genus of $ PSL_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987), pp. 59-86. MR 0916200 (89e:20087)
  • [GT] R.M. Guralnick and J.G. Thompson, Finite Groups of Genus Zero, J. Algebra 131 (1990), pp. 303-341. MR 1055011 (91e:20006)
  • [H] A. Hurwitz, Ueber algebraische gebilde mit eindeutigen transformationen in sich, Math. Ann. 41 (1897) pp. 428-471. MR 1510753
  • [JS] G.A. Jones and D. Singerman, Maps, Hypermaps, and Triangle Groups, in The Grothendieck Theory of Dessins d'Enfants (ed. L. Schneps), London Math. Soc. Lecture Note Ser. 200 (1994), pp. 115-145. MR 1305395 (95m:20055)
  • [K] R. Kulkarni, Symmetries of Surfaces, Topology 26, no. 2 (1987), pp. 195-203. MR 0895571 (88m:57051)
  • [KM] R. Kulkarni and C. Maclachlan, Cyclic $ p$-groups of symmetries of surfaces, Glasgow Math. J. 33 (1991) no. 2, pp. 213-221. MR 1108745 (92e:57052)
  • [Ma1] A.M. MacBeath, Generators of the linear fractional groups, Proc. Symp. Pure Math. 12 (1969), pp. 14-32. MR 0262379 (41:6987)
  • [Ma2] A.M. Macbeath, Hurwitz Groups and Surfaces, in The Eightfold Way, MSRI Publications, Volume 35 (1998), pp. 103-113. MR 1722414 (2001c:14002)
  • [Mi] R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, Vol. 5, American Mathematical Society (1995). MR 1326604 (96f:14029)
  • [MM] B. Malle and B.H. Matzat, Inverse Galois Theory, Springer-Verlag, Berlin (1999). MR 1711577 (2000k:12004)
  • [McM] D. McCullough and A. Miller, The stable genus increment for group actions on closed 2-manifolds, Topology 31 (1992), pp. 367-397. MR 1167177 (93d:57024)
  • [MP] M. Mulase and M. Penkava, Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined Over $ \overline{\mathbb{Q}}$, Asian J. Math. 2 (1998), pp. 875-919. MR 1734132 (2001g:30028)
  • [MS] A. Miller and J.A. Smith, Klein's group and the Hurwitz semigroup, preprint (1991).
  • [Sc] L. Schneps (editor), The Grothendieck Theory of Dessins d'Enfants, London Mathematical Society Lecture Note Series 200, Cambridge University Press, Cambridge (1994). MR 1305390 (95f:11001)
  • [Se] J.P. Serre, Topics in Galois Theory, Research Notes in Mathematics, Vol. 1, Jones and Bartlett Publishers, Boston (1992). MR 1162313 (94d:12006)
  • [SV] G.B. Shabat and V.A. Voevodsky, Drawing Curves Over Number Fields in The Grothendieck Festschrift III, Progress in Math. 88, Birkhäuser, Basel (1990), pp. 199-227. MR 1106916 (92f:11083)
  • [Si] C. Simmons, Euclidean, conformal and hyperbolic geometry over classical, finite and other fields, Ph.D. thesis (1998).
  • [St] R.E. Stong, Pseudofree actions and the Greedy algorithm, Math. Ann. 265 (1983), pp. 501-512. MR 0721884 (85i:57012)
  • [Su] M. Suzuki, Group Theory, Springer-Verlag, Berlin, New York (1982). MR 0648772 (82k:20001c)
  • [T] T. Tucker, Finite groups acting on surfaces and the genus of a group, J. Comb. Theory Ser. B 34, no. 1 (1983), pp. 82-98. MR 0701174 (85b:20055)
  • [Y] K. Yang, Compact Riemann surfaces and algebraic curves, World Scientific, Singapore (1988).MR 0986072 (90e:14023)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M60, 12F12, 20H10, 30F10

Retrieve articles in all journals with MSC (2000): 57M60, 12F12, 20H10, 30F10


Additional Information

Murad Özaydin
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: mozaydin@math.ou.edu

Charlotte Simmons
Affiliation: Department of Marthematics and Statistics, University of Central Oklahoma, Edmond, Oklahoma 73034
Email: cksimmons@ucok.edu

Jennifer Taback
Affiliation: Department of Mathematics, Bowdoin College, Brunswick, Maine 04011
Email: jtaback@bowdoin.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04011-6
Received by editor(s): February 1, 2003
Received by editor(s) in revised form: March 14, 2005
Published electronically: December 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society