Bordered Klein surfaces with maximal symmetry

Greenleaf, Newcomb; May, Coy L.

doi:10.1090/S0002-9947-1982-0670931-X

Bordered Klein surfaces with maximal symmetry
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by Newcomb Greenleaf and Coy L. May PDF

Trans. Amer. Math. Soc. 274 (1982), 265-283 Request permission

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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