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Transactions of the American Mathematical Society

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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
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Abstract: A compact bordered Klein surface of (algebraic) genus $ g \geqslant 2$ is said to have maximal symmetry if its automorphism group is of order $ 12(g - 1)$, the largest possible. In this paper we study the bordered surfaces with maximal symmetry and their automorphism groups, the $ {M^\ast}$-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 $ {M^\ast}$-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 $ {M^\ast}$-simple, that is, if its automorphism group has no proper $ {M^\ast}$-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 $ {M^\ast}$-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, $ {M^\ast}$-group, full covering, boundary degree, $ {M^\ast}$-simple group, primitive maximal symmetry, fundamental group, homology group, projective linear group, regular map
Article copyright: © Copyright 1982 American Mathematical Society

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