Abstract. A compact bordered Klein surface of (algebraic) genus g > 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 A/*-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*-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*-simple, that is, if its automorphism group has no proper M*-quotient group. Our results yield an approach to the problem of classifying the bordered Klein surfaces with maxima] 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*-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. 0. Introduction. In the fundamental paper [6] Hurwitz showed that a compact Riemann surface of genus g > 2 has at most 84(g -1) automorphisms. Recent research [7,8,16] has studied the values of g for which this bound is attained and the structure of the automorphism group in these cases.A compact bordered Klein surface [1] of genus g>2 has at most 12(g-1) automorphisms [10]. In this paper we study the surfaces for which this bound is attained, the bordered surfaces which have "maximal symmetry." We also examine the automorphism groups of these surfaces, the M*-groups [11]. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated Af*-group. §1 contains preliminary results and definitions, while §2 classifies the bordered surfaces with maximal symmetry of low topological genus.§3 is the central section of the paper. Surfaces with maximal symmetry are seen to be "full" coverings of those with "primitive" maximal symmetry. The latter surfaces are those whose automorphism groups are "A/*-simple." The classification problem for surfaces with maximal symmetry then breaks into three parts, which are considered in the next three sections.

) 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

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Let G be a finite group. The strong symmetric genus σ 0 (G) is the minimum genus of any Riemann surface on which G acts, preserving orientation. For any non-negative integer g, there is at least one group of strong symmetric genus g. For g = 2, one such group has the form Z k × D n for some k and n.