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

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:
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