Vol. 111, No. 2, 1984

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The species of bordered Klein surfaces with maximal symmetry of low genus

Coy Lewis May

Vol. 111 (1984), No. 2, 371–394

A compact bordered Klein surface of genus g 2 is said to have maximal symmetry if its automorphism group is of order 12(g 1), the largest possible. For each value of the positive integer g there are, of course, several different topological types of bordered surfaces of genus g; each distinct topological type is called a species of the genus g. Here we classify the species of bordered Klein surfaces with maximal symmetry of genus g 40; there are 32 species in 18 different genera. We also classify the species with maximal symmetry that have no more than 5 boundary components. To aid in the classification two group-theoretic constructions that give new surfaces with maximal symmetry and a family of M-groups are introduced. We also establish several general results about the species of a surface with maximal symmetry. In particular we show that if X is a non-orientable bordered surface with maximal symmetry and solvable automorphism group, then the genus of X is odd.

Mathematical Subject Classification 2000
Primary: 30F20
Secondary: 14H99, 20B25, 57N05
Received: 16 July 1982
Published: 1 April 1984
Coy Lewis May