Bordered Klein surfaces with maximal symmetry
Authors:
Newcomb Greenleaf and Coy L. May
Journal:
Trans. Amer. Math. Soc. 274 (1982), 265283
MSC:
Primary 14H30; Secondary 14E10, 57M10
MathSciNet review:
670931
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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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719820670931X
PII:
S 00029947(1982)0670931X
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
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© Copyright 1982
American Mathematical Society
