Large automorphism groups of compact Klein surfaces with boundary, I

Abstract

) 0. Introduction. Let X be a Klein surface [1], that is, X is a surface with boundary dX together with a dianalytic structure on X. A homeomorphism / : X->X of X onto itself that is dianalytic will be called an automorphism of X.In a recent paper [8] we showed that a compact Klein surface of (algebraic) genus g ^ 2, with non-empty boundary, cannot have more than \2{g-1) automorphisms. We also showed that the bound \2(g-1) is attained, by exhibiting some surfaces of low genus (g = 2, 3, 5) together with their automorphism groups. The corresponding bound for Riemann surfaces is quite well known; Hurwitz showed that a compact Riemann surface of genus g ^ 2 has at most 84(#-1) (orientation-preserving) automorphisms.Here we study those groups that act as a group of \2{g -\) automorphisms of a compact Klein surface with boundary of genus g ^ 2. Our main result is a characterization of these groups in terms of their presentations. We call these finite groups M*-groups. It is easy to find examples of M*-groups. In fact, by using known results about normal subgroups of the modular group, we are able to find an infinite family of A/*-groups. Consequently the bound \2{g-1) is attained for infinitely many values of the genus g.In the final section of the paper we show how to obtain other infinite families for which the bound \2{g-lj is achieved, without use of results about the modular group. To obtain an infinite family we only need a single Klein surface with boundary that has \2{g-\) automorphisms. Some of the infinite families we get in this manner consist of orientable surfaces; others consist of non-orientable surfaces. Compact Klein surfaces and NEC groups.Let A' be a compact Klein surface, and let E be the field of all meromorphic functions on X. E is an algebraic function field in one variable over R [1, p. 102] and as such has an algebraic genus g. This is the non-negative integer that makes the algebraic version of the Riemann-Roch Theorem work [2, p. 22]. We will refer to g simply as the genus of the compact Klein surface X. In case Xh a Riemann surface, g is equal to the topological genus of X. Now let (X c , n, a) be the complex double of X [1, pp. 37-41], that is, X c is a compact Riemann surface, n: X c -> X is an unramified 2-sheeted covering of X, and a is the unique antianalytic involution of X c such that n = n°o. If F is the field of meromorphic functions on X c) then F = E(i) and by a well-known classical result [2, p. 99], the genus of X is equal to the genus of X c .If

Symmetry types of hyperelliptic Riemann surfaces

“…We now summarize some general results concerning automorphisms of Klein surfaces. The full group of dianalytic automorphisms of the Klein surface K will be denoted by Aut ± K. 38]). -Let Aut ± K be the full group of automorphisms of the Klein surface K = H/Γ and let N (Γ) be the normalizer of Γ in Aut ± H. Then…”

Complex doubles of bordered Klein surfaces with maximal symmetry

“…A group that acts as the automorphism group of a bordered surface with maximal symmetry is called an M*-group [8]. The first important result about Af*-groups was that they must have a certain partial presentation [8, p. 5].…”

Nilpotent automorphism groups of bordered Klein surfaces

“…The Riemann-Hurwitz formula was used in [3] to establish the general bound 12(g -1) for the order of an automorphism group. If the order of the group is large enough, then the group's action is also determined.…”

“…Let A" be a compact bordered Klein surface of algebraic genus g > 2. Then the automorphism group G of X is finite, and the order of G is at most 12(g -1) [3]. If the order of G is the largest possible, then G is called an M*-group.…”