Bordered Klein surfaces with maximal symmetry

Authors:
Newcomb Greenleaf and Coy L. May

Journal:
Trans. Amer. Math. Soc. **274** (1982), 265-283

MSC:
Primary 14H30; Secondary 14E10, 57M10

DOI:
https://doi.org/10.1090/S0002-9947-1982-0670931-X

MathSciNet review:
670931

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A compact bordered Klein surface of (algebraic) genus is said to have *maximal symmetry* if its automorphism group is of order , the largest possible. In this paper we study the bordered surfaces with maximal symmetry and their automorphism groups, the -groups. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated -group. We begin by classifying the bordered surfaces with maximal symmetry of low topological genus. We next show that a bordered surface with maximal symmetry is a full covering of another surface with *primitive* maximal symmetry. A surface has primitive maximal symmetry if its automorphism group is *-simple*, that is, if its automorphism group has no proper -quotient group. Our results yield an approach to the problem of classifying the bordered Klein surfaces with maximal symmetry. Next we obtain several constructions of full covers of a bordered surface. These constructions give numerous infinite families of surfaces with maximal symmetry. We also prove that only two of the -simple groups are solvable, and we exhibit infinitely many nonsolvable ones. Finally we show that there is a correspondence between bordered Klein surfaces with maximal symmetry and regular triangulations of surfaces.

**[1]**N. L. Ailing and N. Greenleaf,*Foundations of the theory of Klein surfaces*, Lecture Notes in Math., Vol. 219, Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR**0333163 (48:11488)****[2]**H. S. M. Coxeter,*The abstract groups*, Trans. Amer. Math. Soc.**45**(1939), 73-150. MR**1501984****[3]**H. S. M. Coxeter and W. O. J. Moser,*Generators and relations for discrete groups*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, Berlin-Heidelberg-New York, 1972. MR**0349820 (50:2313)****[4]**W. Hall,*Automorphisms and coverings of Klein surfaces*, Ph. D. Thesis, University of Southampton, 1977.**[5]**M. Heins,*On the number of**directly conformal maps which a multiply-connected plane region of finite connectivity**admits onto itself*, Bull. Amer. Math. Soc.**52**(1946), 454-457. MR**0016469 (8:21d)****[6]**A. Hurwitz,*Uber algebraische Gebilde mit eindeutigen Transformationen in sich*, Math. Ann.**41**(1893), 403-442. MR**1510753****[7]**A. M. Macbeath,*On a theorem of Hurwitz*, Proc. Glasgow Math. Assoc.**5**(1961), 90-96. MR**0146724 (26:4244)****[8]**-,*Generators of the linear fractional groups*, Number Theory (Houston, 1967), Proc. Sympos. Pure Math., Vol. 12, Amer. Math. Soc., Providence, R.I., 1969, pp. 14-32. MR**0262379 (41:6987)****[9]**W. S. Massey,*Algebraic topology: An introduction*, Harcourt, Brace & World, New York, 1967. MR**0211390 (35:2271)****[10]**C. L. May,*Automorphisms of compact Klein surfaces with boundary*, Pacific J. Math.**59**(1975), 199-210. MR**0399451 (53:3295)****[11]**-,*Large automorphism groups of compact Klein surfaces with boundary*, Glasgow Math. J.**18**(1977), 1-10. MR**0425113 (54:13071)****[12]**-,*A bound for the number of automorphisms of a compact Klein surface with boundary*, Proc. Amer. Math. Soc.**63**(1977), 273-280. MR**0435385 (55:8345)****[13]**-,*Cyclic automorphism groups of compact bordered Klein surfaces*, Houston J. Math.**3**(1977), 395-405. MR**0457710 (56:15914)****[14]**K. Oikawa,*Notes on conformal mappings of a Riemann surface onto itself*, Kodai 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**0080730 (18:290d)****[15]**J. Rotman,*The theory of groups*, Allyn and Bacon, Boston, Mass., 1965. MR**0204499 (34:4338)****[16]**C. H. Sah,*Groups related to compact Riemann surfaces*, Acta Math.**123**(1969), 13-42. MR**0251216 (40:4447)****[17]**F. A. Sherk,*The regular maps on a surface of genus three*, Canad. J. Math.**11**(1959), 452-480. MR**0107203 (21:5928)****[18]**D. Singerman,*Symmetries of Riemann surfaces with large automorphism group*, Math. Ann.**210**(1974), 17-32. MR**0361059 (50:13505)****[19]**H. C. Wilkie,*On non-euclidean crystallographic groups*, Math. Z.**91**(1966), 87-102. MR**0185013 (32:2483)****[20]**S. E. Wilson,*Riemann surfaces over regular maps*, Canad. J. Math.**30**(1978), 763-782. MR**0493918 (58:12876)****[21]**-,*Operators over regular maps*, Pacific J. Math.**81**(1979), 559-568. MR**547621 (81a:57004)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14H30,
14E10,
57M10

Retrieve articles in all journals with MSC: 14H30, 14E10, 57M10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0670931-X

Keywords:
Bordered Klein surface,
genus,
automorphism,
maximal symmetry,
-group,
full covering,
boundary degree,
-simple group,
primitive maximal symmetry,
fundamental group,
homology group,
projective linear group,
regular map

Article copyright:
© Copyright 1982
American Mathematical Society